P. 386 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 38501-38600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	ovnsubadd 38501*	voln^* is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
voln^* Σ^{^} voln^*

Theorem	ovnome 38502	voln^* is an outer measure on the space of multidimensional real numbers with dimension equal to the cardinality of the finite set . Proposition 115D (a) of [Fremlin1] p. 30 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
voln^* OutMeas

Theorem	vonmea 38503	voln is a measure on the space of multidimensional real numbers with dimension equal to the cardinality of the finite set . Comments in Definition 115E of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 11-Oct-2020.)
voln Meas

Theorem	volicon0 38504	The measure of a nonempty left-closed, right-open interval. (Contributed by Glauco Siliprandi, 21-Nov-2020.)


Theorem	hsphoif 38505*	is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)


Theorem	hoidmvval 38506*	The dimensional volume of a multidimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)


Theorem	hoissrrn2 38507*	A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$/\$ $/\$

Theorem	hsphoival 38508*	is a function (that returns the representation of the right side of a half-open interval intersected with a half-space). Step (b) in Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)


Theorem	hoiprodcl3 38509*	The pre-measure of half-open intervals is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$/\$ $/\$

Theorem	volicore 38510	The Lebesgue measure of a left-closed right-open interval is a real number. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$/\$

Theorem	hoidmvcl 38511*	The dimensional volume of a multidimensional half-open interval is a nonnegative real. (Contributed by Glauco Siliprandi, 21-Nov-2020.)


Theorem	hoidmv0val 38512*	The dimensional volume of a 0-dimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)


Theorem	hoidmvn0val 38513*	The dimensional volume of a non 0-dimensional half-open interval. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)


Theorem	hsphoidmvle2 38514*	The dimensional volume of a half-open interval intersected with a two half-spaces. Used in the last inequality of step (c) of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$\$

Theorem	hsphoidmvle 38515*	The dimensional volume of a half-open interval intersected with a half-space, is less than or equal to the dimensional volume of the original half-open interval. Used in the last inequality of step (e) of Lemma 115B of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$\$

Theorem	hoidmvval0 38516*	The dimensional volume of the (half-open interval) empty set. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)


Theorem	hoiprodp1 38517*	The dimensional volume of a half-open interval with dimension . Used in the first equality of step (e) of Lemma 115B of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)


Theorem	sge0hsphoire 38518*	If the generalized sum of dimensional volumes of n-dimensional half-open intervals is finite, then the sum stays finite if every half-open interval is intersected with a half-space. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$\$ Σ^{^} Σ^{^}

Theorem	hoidmvval0b 38519*	The dimensional volume of the (half-open interval) empty set. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)


Theorem	hoidmv1lelem1 38520*	The supremum of belongs to . This is the last part of step (a) and the whole step (b) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^{^} Σ^{^} $/\$ $/\$

Theorem	hoidmv1lelem2 38521*	This is the contradiction proven in step (c) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^{^} Σ^{^}

Theorem	hoidmv1lelem3 38522*	The dimensional volume of a 1-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. This is the non-empty, finite generalized sum, sub case in Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^{^} Σ^{^} Σ^{^}

Theorem	hoidmv1le 38523*	The dimensional volume of a 1-dimensional half-open interval is less than or equal to the generalized sum of the dimensional volumes of countable half-open intervals that cover it. This is one of the two base cases of the induction of Lemma 115B of [Fremlin1] p. 29 (the other base case is the 0-dimensional case). This proof of the 1-dimensional case is given in Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^{^}

Theorem	hoidmvlelem1 38524*	The supremum of belongs to . Step (c) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$\$ Σ^{^} Σ^{^}

Theorem	hoidmvlelem2 38525*	This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$\$ Σ^{^} Σ^{^} inf

Theorem	hoidmvlelem3 38526*	This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$\$ $/\$ Σ^{^} Σ^{^} Σ^{^}

Theorem	hoidmvlelem4 38527*	The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step of Lemma 115B of [Fremlin1] p. 29, case nonempty interval and dimension of the space greater than . (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$\$ $/\$ Σ^{^} Σ^{^} Σ^{^} Σ^{^}

Theorem	hoidmvlelem5 38528*	The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$\$ Σ^{^} Σ^{^}

Theorem	hoidmvle 38529*	The dimensional volume of a n-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Σ^{^}

Theorem	ovnhoilem1 38530*	The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. First part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$/\$ Σ^{^} voln^*

Theorem	ovnhoilem2 38531*	The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. Second part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
$/\$ Σ^{^} voln^*

Theorem	ovnhoi 38532*	The Lebesgue outer measure of a multidimensional half-open interval is its dimensional volume (the product of its length in each dimension, when the dimension is nonzero). Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
voln^*

Theorem	dmovn 38533	The domain of the Lebesgue outer measure is the power set of the n-dimensional Real numbers. Step (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30 (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln^*

Theorem	hoicoto2 38534*	The half-open interval expressed using a composition of a function into and using two distinct real valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020.)


Theorem	dmvon 38535	Lebesgue measurable n-dimensional subsets of Reals. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln CaraGenvoln^*

Theorem	hoi2toco 38536*	The half-open interval expressed using a composition of a function into and using two distinct real valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020.)


Theorem	hoidifhspval 38537*	is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)


Theorem	hspval 38538*	The value of the half-space of n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)


Theorem	ovnlecvr2 38539*	Given a subset of multidimensional reals and a set of half-open intervals that covers it, the Lebesgue outer measure of the set is bounded by the generalized sum of the pre-measure of the half-open intervals. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln^* Σ^{^}

Theorem	ovncvr2 38540*	and are the left and right side of a cover of . This cover is made of n-dimensional half open intervals, and approximates the n-dimensional Lebesgue outer volume of . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Σ^{^} voln^* $/\$ $/\$ $/\$ Σ^{^} voln^*

Theorem	dmovnsal 38541	The domain of the Lebesgue measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln SAlg

Theorem	unidmovn 38542	Base set of the n-dimensional Lebesgue outer measure (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln^*

Theorem	rrnmbl 38543	The set of n-dimensional Real numbers is Lebesgue measurable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln

Theorem	hoidifhspval2 38544*	is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)


Theorem	hspdifhsp 38545*	A n-dimensional half-open interval is the intersection of the difference of half spaces. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
$\$

Theorem	unidmvon 38546	Base set of the n-dimensional Lebesgue measure. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln

Theorem	hoidifhspf 38547*	is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)


Theorem	hoidifhspval3 38548*	is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)


Theorem	hoidifhspdmvle 38549*	The dimensional volume of the difference of a half-open interval and a half-space is less than or equal to the dimensional volume of the whole half-open interval. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)


Theorem	voncmpl 38550	The Lebesgue measure is complete. See Definition 112Df of [Fremlin1] p. 19. This is an observation written after Definition 115E of [Fremlin1] p. 31 (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln voln voln

Theorem	hoiqssbllem1 38551*	The center of the n-dimensional ball belongs to the half-open interval. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
$/\$ $/\$

Theorem	hoiqssbllem2 38552*	The center of the n-dimensional ball belongs to the half-open interval. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
$/\$ $/\$ ℝ^

Theorem	hoiqssbllem3 38553*	A n-dimensional ball contains a non-empty half-open interval with vertices with rational components. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
$/\$ ℝ^

Theorem	hoiqssbl 38554*	A n-dimensional ball contains a non-empty half-open interval with vertices with rational components. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
$/\$ ℝ^

Theorem	hspmbllem1 38555*	Any half-space of the n-dimensional Real numbers is Lebesgue measurable. This is Step (a) of Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
$\$

Theorem	hspmbllem2 38556*	Any half-space of the n-dimensional Real numbers is Lebesgue measurable. This is Step (b) of Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Σ^{^} voln^* voln^* voln^* voln^* $\$ $\$ voln^* voln^* $\$ voln^*

Theorem	hspmbllem3 38557*	Any half-space of the n-dimensional Real numbers is Lebesgue measurable. Lemma 115F of [Fremlin1] p. 31. This proof handles the non-trivial cases (non-zero dimension and finite outer measure) (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln^* Σ^{^} voln^* voln^* voln^* $\$ voln^*

Theorem	hspmbl 38558*	Any half-space of the n-dimensional Real numbers is Lebesgue measurable. Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln

Theorem	hoimbllem 38559*	Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln

Theorem	hoimbl 38560*	Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln

Theorem	opnvonmbllem1 38561*	The half-open interval expressed using a composition of a function (Contributed by Glauco Siliprandi, 24-Dec-2020.)


Theorem	opnvonmbllem2 38562*	An open subset of the n-dimensional Real numbers is Lebesgue measurable. This is Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln ℝ^

Theorem	opnvonmbl 38563	An open subset of the n-dimensional Real numbers is Lebesgue measurable. This is Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
voln ℝ^

Theorem	opnssborel 38564	Open sets of a generalized real Euclidean space are Borel sets (notice that this theorem is even more general, because is not required to be a set). (Contributed by Glauco Siliprandi, 3-Jan-2021.)
ℝ^ SalGen

Theorem	borelmbl 38565	All Borel subsets of the n-dimensional Real numbers are Lebesgue measurable. This is Proposition 115G (b) of [Fremlin1] p. 32. See also Definition 111G (d) of [Fremlin1] p. 13. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
voln SalGenℝ^

Theorem	volicorege0 38566	The Lebesgue measure of a left-closed right-open interval with real bounds, is a nonnegative real number. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$

Theorem	isvonmbl 38567*	The predicate " is measurable w.r.t. the n-dimensional Lebesgue measure". A set is measurable if it splits every other set in a "nice" way, that is, if the measure of the pieces and $\$ sum up to the measure of . Definition 114E of [Fremlin1] p. 25. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln $/\$ voln^* voln^* $\$ voln^*

Theorem	mblvon 38568	The n-dimensional Lebesgue measure of a measurable set is the same as its n-dimensional Lebesgue outer measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln voln voln^*

Theorem	vonmblss 38569	n-dimensional Lebesgue measurable sets are subsets of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln

Theorem	volico2 38570	The measure of left closed, right open interval. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$

Theorem	vonmblss2 38571	n-dimensional Lebesgue measurable sets are subsets of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln

Theorem	ovolval2lem 38572*	The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^}. (Contributed by Glauco Siliprandi, 3-Mar-2021.)


Theorem	ovolval2 38573*	The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^}. See ovolval 22481 for an alternative expression. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$ Σ^{^} inf

Theorem	ovnsubadd2lem 38574*	voln^* is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . The special case of the union of 2 sets. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln^* voln^voln^

Theorem	ovnsubadd2 38575	voln^* is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . The special case of the union of 2 sets. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln^* voln^voln^

Theorem	ovolval3 38576*	The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^} and . See ovolval 22481 and ovolval2 38573 for alternative expressions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$ Σ^{^} inf

Theorem	ovnsplit 38577	The n-dimensional Lebesgue outer measure function is finitely sub-additive: application to a set split in two parts. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln^* voln^* voln^* $\$

Theorem	ovolval4lem1 38578*	\|- ( ( ph /\ n e. A ) -> ( ( (,) o. G ) n ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$

Theorem	ovolval4lem2 38579*	The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 38576, but here is may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$ Σ^{^} inf

Theorem	ovolval4 38580*	The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 38576, but here may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$ Σ^{^} inf

Theorem	ovolval5lem1 38581*	\|- ( ph -> ( sum^ ( ( A - ( W / ( 2 ^ n ) ) ) (,) B ) ) ) ) <_ ( ( sum^ ( A [,) B ) ) ) ) +e W ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$ $/\$ Σ^{^} Σ^{^}

Theorem	ovolval5lem2 38582*	\|- ( ( ph /\ n e. NN ) -> <. ( ( 1st n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd n ) ) >. e. ( RR X. RR ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$ Σ^{^} Σ^{^} Σ^{^}

Theorem	ovolval5lem3 38583*	The value of the Lebesgue outer measure for subsets of the reals, using covers of left-closed right-open intervals are used, instead of open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$ Σ^{^} $/\$ Σ^{^} inf inf

Theorem	ovolval5 38584*	The value of the Lebesgue outer measure for subsets of the reals, using covers of left-closed right-open intervals are used, instead of open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$ Σ^{^} inf

Theorem	ovnovollem1 38585*	if is a cover of in , then is the corresponding cover in the space of 1-dimensional reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Σ^{^} $/\$ Σ^{^}

Theorem	ovnovollem2 38586*	if is a cover of in ℝ^, then is the corresponding cover in the reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Σ^{^} $/\$ Σ^{^}

Theorem	ovnovollem3 38587*	The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$/\$ Σ^{^} $/\$ Σ^{^} voln^*

Theorem	ovnovol 38588	The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln^*

Theorem	vonvolmbllem 38589*	If a subset of real numbers is Lebesgue measurable, then its corresponding 1-dimensional set is measurable w.r.t. the n-dimensional Lebesgue measure, (with equal to ). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
$\$ voln^* voln^* $\$ voln^*

Theorem	vonvolmbl 38590	A subset of Real numbers is Lebesgue measurable if and only if its corresponding 1-dimensional set is measurable w.r.t. the 1-dimensional Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln

Theorem	vonvol 38591	The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln

Theorem	vonvolmbl2 38592*	A subset of the space of 1-dimensional Real numbers is Lebesgue measurable if and only if its projection on the Real numbers is Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln

Theorem	vonvol2 38593*	The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
voln voln

21.31 Mathbox for Saveliy Skresanov

21.31.1 Ceva's theorem

Theorem	sigarval 38594*	Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
$/\$

Theorem	sigarim 38595*	Signed area takes value in reals. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
$/\$

Theorem	sigarac 38596*	Signed area is anticommutative. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
$/\$

Theorem	sigaraf 38597*	Signed area is additive by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
$/\$ $/\$

Theorem	sigarmf 38598*	Signed area is additive (with respect to subtraction) by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
$/\$ $/\$

Theorem	sigaras 38599*	Signed area is additive by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
$/\$ $/\$

Theorem	sigarms 38600*	Signed area is additive (with respect to subtraction) by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
$/\$ $/\$

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7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-40909

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