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**Theorem List for Metamath Proof Explorer -** 33701-33800 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	osumcllem6N 33701	Lemma for osumclN 33707. Use atom exchange hlatexch1 33135 to swap and . (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	osumcllem7N 33702*	Lemma for osumclN 33707. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem8N 33703	Lemma for osumclN 33707. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem9N 33704	Lemma for osumclN 33707. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem10N 33705	Lemma for osumclN 33707. Contradict osumcllem9N 33704. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	osumcllem11N 33706	Lemma for osumclN 33707. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	osumclN 33707	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 33708	For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 33592 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidN 33709	Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 33693. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 33707. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	pexmidlem1N 33710	Lemma for pexmidN 33709. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$

Theorem	pexmidlem2N 33711	Lemma for pexmidN 33709. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidlem3N 33712	Lemma for pexmidN 33709. Use atom exchange hlatexch1 33135 to swap and . (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	pexmidlem4N 33713*	Lemma for pexmidN 33709. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem5N 33714	Lemma for pexmidN 33709. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem6N 33715	Lemma for pexmidN 33709. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem7N 33716	Lemma for pexmidN 33709. Contradict pexmidlem6N 33715. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pexmidlem8N 33717	Lemma for pexmidN 33709. The contradiction of pexmidlem6N 33715 and pexmidlem7N 33716 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 33718	Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 33693. 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 33719	Lemma for pl42N 33723. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	pl42lem2N 33720	Lemma for pl42N 33723. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	pl42lem3N 33721	Lemma for pl42N 33723. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pl42lem4N 33722	Lemma for pl42N 33723. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$

Theorem	pl42N 33723	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 33724	Extend class notation with set of all co-atoms (lattice hyperplanes).


Syntax	claut 33725	Extend class notation with set of all lattice automorphisms.


Syntax	cwpointsN 33726	Extend class notation with W points.


Syntax	cpautN 33727	Extend class notation with set of all projective automorphisms.


Definition	df-lhyp 33728*	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 33729*	Define set of lattice autoisomorphisms. (Contributed by NM, 11-May-2012.)
$/\$

Definition	df-watsN 33730*	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 33731*	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 33732*	The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$\$

Theorem	watvalN 33733	Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$ $\$

Theorem	iswatN 33734	The predicate "is a W atom" (corresponding to fiducial atom ). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
$/\$ $/\$

Theorem	lhpset 33735*	The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)


Theorem	islhp 33736	The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 11-May-2012.)
$/\$

Theorem	islhp2 33737	The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 18-May-2012.)
$/\$

Theorem	lhpbase 33738	A co-atom is a member of the lattice base set (i.e. a lattice element). (Contributed by NM, 18-May-2012.)


Theorem	lhp1cvr 33739	The lattice unit covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.)
$/\$

Theorem	lhplt 33740	An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$ $/\$ $/\$

Theorem	lhp2lt 33741	The join of two atoms under a co-atom is strictly less than it. (Contributed by NM, 8-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhpexlt 33742*	There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$

Theorem	lhp0lt 33743	A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
$/\$

Theorem	lhpn0 33744	A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.)
$/\$

Theorem	lhpexle 33745*	There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013.)
$/\$

Theorem	lhpexnle 33746*	There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.)
$/\$

Theorem	lhpexle1lem 33747*	Lemma for lhpexle1 33748 and others that eliminates restrictions on . (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lhpexle1 33748*	There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$

Theorem	lhpexle2lem 33749*	Lemma for lhpexle2 33750. (Contributed by NM, 19-Jun-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	lhpexle2 33750*	There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$ $/\$

Theorem	lhpexle3lem 33751*	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 33752*	There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	lhpex2leN 33753*	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 33754	The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.)
$/\$

Theorem	lhpoc2N 33755	The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
$/\$

Theorem	lhpocnle 33756	The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)
$/\$

Theorem	lhpocat 33757	The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.)
$/\$

Theorem	lhpocnel 33758	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 33759	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 33760	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 33761	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 33762	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 33763	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 33764*	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 33765	Specialization of lhpmcvr2 33764. TODO: Use this to simplify many uses of $.\/$ $./\$ to become . (Contributed by NM, 6-Apr-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpmcvr4N 33766	Specialization of lhpmcvr2 33764. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$

Theorem	lhpmcvr5N 33767*	Specialization of lhpmcvr2 33764. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpmcvr6N 33768*	Specialization of lhpmcvr2 33764. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$

Theorem	lhpm0atN 33769	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 33770	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 33771	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 33772	Join and meet with different atoms under co-atom . (Contributed by NM, 15-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhp2atnle 33773	Inequality for 2 different atoms under co-atom . (Contributed by NM, 17-Jun-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhp2atne 33774	Inequality for joins with 2 different atoms under co-atom . (Contributed by NM, 22-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	lhp2at0nle 33775	Inequality for 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $.\/$

Theorem	lhp2at0ne 33776	Inequality for joins with 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)
$.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Theorem	lhpelim 33777	Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 33770 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $./\$

Theorem	lhpmod2i2 33778	Modular law for hyperplanes analogous to atmod2i2 33602 for atoms. (Contributed by NM, 9-Feb-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$

Theorem	lhpmod6i1 33779	Modular law for hyperplanes analogous to complement of atmod2i1 33601 for atoms. (Contributed by NM, 1-Jun-2013.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Theorem	lhprelat3N 33780*	The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 33152. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
$./\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$

Theorem	cdlemb2 33781*	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 33782	Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpat 33783	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 33784	Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)
$.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$

Theorem	lhpat2 33785	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 33786	There is only one atom under both $.\/$ and co-atom . (Contributed by NM, 21-Nov-2012.)
$.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Theorem	4atexlemk 33787	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemw 33788	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlempw 33789	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$

Theorem	4atexlemp 33790	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemq 33791	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlems 33792	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemt 33793	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemutvt 33794	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

Theorem	4atexlempnq 33795	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemnslpq 33796	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Theorem	4atexlemkl 33797	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemkc 33798	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlemwb 33799	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$

Theorem	4atexlempsb 33800	Lemma for 4atexlem7 33815. (Contributed by NM, 23-Nov-2012.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$

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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 35701-35713

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