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**Theorem List for Metamath Proof Explorer -** 33501-33600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	osumclN 33501	Closure of orthogonal sum. If and are orthogonal closed projective subspaces, then their sum is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	pmapojoinN 33502	For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 33386 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidN 33503	Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 33487. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 33501. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pexmidlem1N 33504	Lemma for pexmidN 33503. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$

Theorem	pexmidlem2N 33505	Lemma for pexmidN 33503. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidlem3N 33506	Lemma for pexmidN 33503. Use atom exchange hlatexch1 32929 to swap and . (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidlem4N 33507*	Lemma for pexmidN 33503. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem5N 33508	Lemma for pexmidN 33503. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem6N 33509	Lemma for pexmidN 33503. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem7N 33510	Lemma for pexmidN 33503. Contradict pexmidlem6N 33509. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem8N 33511	Lemma for pexmidN 33503. The contradiction of pexmidlem6N 33509 and pexmidlem7N 33510 shows that there can be no atom that is not in , which is therefore the whole atom space. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	pexmidALTN 33512	Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 33487. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables , , , , in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pl42lem1N 33513	Lemma for pl42N 33517. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	pl42lem2N 33514	Lemma for pl42N 33517. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	pl42lem3N 33515	Lemma for pl42N 33517. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pl42lem4N 33516	Lemma for pl42N 33517. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	pl42N 33517	Law holding in a Hilbert lattice that fails in orthomodular lattice L42 (Figure 7 in [MegPav2000] p. 2366). (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Syntax	clh 33518	Extend class notation with set of all co-atoms (lattice hyperplanes).


Syntax	claut 33519	Extend class notation with set of all lattice automorphisms.


Syntax	cwpointsN 33520	Extend class notation with W points.


Syntax	cpautN 33521	Extend class notation with set of all projective automorphisms.


Definition	df-lhyp 33522*	Define the set of lattice hyperplanes, which are all lattice elements covered by 1 (i.e. all co-atoms). We call them "hyperplanes" instead of "co-atoms" in analogy with projective geometry hyperplanes. (Contributed by NM, 11-May-2012.)


Definition	df-laut 33523*	Define set of lattice autoisomorphisms. (Contributed by NM, 11-May-2012.)
$/\$

Definition	df-watsN 33524*	Define W-atoms corresponding to an arbitrary "fiducial (i.e. reference) atom" . These are all atoms not in the polarity of , which is the hyperplane determined by . Definition of set W in [Crawley] p. 111. (Contributed by NM, 26-Jan-2012.)
$\$

Definition	df-pautN 33525*	Define set of all projective automorphisms. This is the intended definition of automorphism in [Crawley] p. 112. (Contributed by NM, 26-Jan-2012.)
$/\$

Theorem	watfvalN 33526*	The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$\$

Theorem	watvalN 33527	Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$ $\$

Theorem	iswatN 33528	The predicate "is a W atom" (corresponding to fiducial atom ). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	lhpset 33529*	The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)


Theorem	islhp 33530	The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 11-May-2012.)
$/\$

Theorem	islhp2 33531	The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 18-May-2012.)
$/\$

Theorem	lhpbase 33532	A co-atom is a member of the lattice base set (i.e. a lattice element). (Contributed by NM, 18-May-2012.)


Theorem	lhp1cvr 33533	The lattice unit covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.)
$/\$

Theorem	lhplt 33534	An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	lhp2lt 33535	The join of two atoms under a co-atom is strictly less than it. (Contributed by NM, 8-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhpexlt 33536*	There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$

Theorem	lhp0lt 33537	A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$

Theorem	lhpn0 33538	A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.)
$/\$

Theorem	lhpexle 33539*	There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013.)
$/\$

Theorem	lhpexnle 33540*	There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.)
$/\$

Theorem	lhpexle1lem 33541*	Lemma for lhpexle1 33542 and others that eliminates restrictions on . (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lhpexle1 33542*	There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$

Theorem	lhpexle2lem 33543*	Lemma for lhpexle2 33544. (Contributed by NM, 19-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lhpexle2 33544*	There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$ $/\$

Theorem	lhpexle3lem 33545*	There exists atom under a co-atom different from any 3 other atoms. TODO: study if adant,simp usage can be improved. (Contributed by NM, 9-Jul-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lhpexle3 33546*	There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	lhpex2leN 33547*	There exist at least two different atoms under a co-atom. This allows us to create a line under the co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	lhpoc 33548	The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.)
$/\$

Theorem	lhpoc2N 33549	The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
$/\$

Theorem	lhpocnle 33550	The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)
$/\$

Theorem	lhpocat 33551	The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.)
$/\$

Theorem	lhpocnel 33552	The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 25-May-2012.)
$/\$ $/\$

Theorem	lhpocnel2 33553	The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 20-Feb-2014.)
$/\$ $/\$

Theorem	lhpjat1 33554	The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhpjat2 33555	The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 4-Jun-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhpj1 33556	The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhpmcvr 33557	The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	lhpmcvr2 33558*	Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpmcvr3 33559	Specialization of lhpmcvr2 33558. TODO: Use this to simplify many uses of $.\/$ $./\$ to become . (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpmcvr4N 33560	Specialization of lhpmcvr2 33558. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$

Theorem	lhpmcvr5N 33561*	Specialization of lhpmcvr2 33558. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpmcvr6N 33562*	Specialization of lhpmcvr2 33558. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$

Theorem	lhpm0atN 33563	If the meet of a lattice hyperplane with a nonzero element is zero, the element is an atom. (Contributed by NM, 28-Apr-2014.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $/\$ $./\$

Theorem	lhpmat 33564	An element covered by the lattice unit, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.)
$./\$ $/\$ $/\$ $/\$ $./\$

Theorem	lhpmatb 33565	An element covered by the lattice unit, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013.)
$./\$ $/\$ $/\$ $./\$

Theorem	lhp2at0 33566	Join and meet with different atoms under co-atom . (Contributed by NM, 15-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhp2atnle 33567	Inequality for 2 different atoms under co-atom . (Contributed by NM, 17-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhp2atne 33568	Inequality for joins with 2 different atoms under co-atom . (Contributed by NM, 22-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	lhp2at0nle 33569	Inequality for 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhp2at0ne 33570	Inequality for joins with 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	lhpelim 33571	Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 33564 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $./\$

Theorem	lhpmod2i2 33572	Modular law for hyperplanes analogous to atmod2i2 33396 for atoms. (Contributed by NM, 9-Feb-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$

Theorem	lhpmod6i1 33573	Modular law for hyperplanes analogous to complement of atmod2i1 33395 for atoms. (Contributed by NM, 1-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	lhprelat3N 33574*	The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 32946. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$

Theorem	cdlemb2 33575*	Given two atoms not under the fiducial (reference) co-atom , there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 30-May-2012.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhple 33576	Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpat 33577	Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpat4N 33578	Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpat2 33579	Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lhpat3 33580	There is only one atom under both $.\/$ and co-atom . (Contributed by NM, 21-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	4atexlemk 33581	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemw 33582	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlempw 33583	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$

Theorem	4atexlemp 33584	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemq 33585	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlems 33586	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemt 33587	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemutvt 33588	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlempnq 33589	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemnslpq 33590	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemkl 33591	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemkc 33592	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemwb 33593	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlempsb 33594	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlemqtb 33595	Lemma for 4atexlem7 33609. (Contributed by NM, 24-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlempns 33596	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemswapqr 33597	Lemma for 4atexlem7 33609. Swap and , so that theorems involving can be reused for . Note that must be expanded because it involves . (Contributed by NM, 25-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemu 33598	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemv 33599	Lemma for 4atexlem7 33609. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	4atexlemunv 33600	Lemma for 4atexlem7 33609. (Contributed by NM, 21-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$

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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 40101-40161

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