:: YELLOW11 semantic presentation
theorem Th1: :: YELLOW11:1
theorem Th2: :: YELLOW11:2
theorem Th3: :: YELLOW11:3
theorem Th4: :: YELLOW11:4
Lemma5:
3 \ 2 c= 3 \ 1
theorem Th5: :: YELLOW11:5
theorem Th6: :: YELLOW11:6
theorem Th7: :: YELLOW11:7
theorem Th8: :: YELLOW11:8
:: deftheorem Def1 defines N_5 YELLOW11:def 1 :
:: deftheorem Def2 defines M_3 YELLOW11:def 2 :
theorem Th9: :: YELLOW11:9
for b
1 being
LATTICE holds
( not ( ex b
2 being
full Sublattice of b
1 st
N_5 ,b
2 are_isomorphic & ( for b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
4 <> b
5 & b
4 <> b
6 & b
5 <> b
6 & b
2 "/\" b
3 = b
2 & b
2 "/\" b
4 = b
2 & b
4 "/\" b
6 = b
4 & b
5 "/\" b
6 = b
5 & b
3 "/\" b
4 = b
2 & b
3 "/\" b
5 = b
3 & b
4 "/\" b
5 = b
2 & b
3 "\/" b
4 = b
6 & b
4 "\/" b
5 = b
6 ) ) ) & not ( ex b
2, b
3, b
4, b
5, b
6 being
Element of b
1 st
( b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
4 <> b
5 & b
4 <> b
6 & b
5 <> b
6 & b
2 "/\" b
3 = b
2 & b
2 "/\" b
4 = b
2 & b
4 "/\" b
6 = b
4 & b
5 "/\" b
6 = b
5 & b
3 "/\" b
4 = b
2 & b
3 "/\" b
5 = b
3 & b
4 "/\" b
5 = b
2 & b
3 "\/" b
4 = b
6 & b
4 "\/" b
5 = b
6 ) & ( for b
2 being
full Sublattice of b
1 holds
not
N_5 ,b
2 are_isomorphic ) ) )
theorem Th10: :: YELLOW11:10
for b
1 being
LATTICE holds
( not ( ex b
2 being
full Sublattice of b
1 st
M_3 ,b
2 are_isomorphic & ( for b
2, b
3, b
4, b
5, b
6 being
Element of b
1 holds
not ( b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
4 <> b
5 & b
4 <> b
6 & b
5 <> b
6 & b
2 "/\" b
3 = b
2 & b
2 "/\" b
4 = b
2 & b
2 "/\" b
5 = b
2 & b
3 "/\" b
6 = b
3 & b
4 "/\" b
6 = b
4 & b
5 "/\" b
6 = b
5 & b
3 "/\" b
4 = b
2 & b
3 "/\" b
5 = b
2 & b
4 "/\" b
5 = b
2 & b
3 "\/" b
4 = b
6 & b
3 "\/" b
5 = b
6 & b
4 "\/" b
5 = b
6 ) ) ) & not ( ex b
2, b
3, b
4, b
5, b
6 being
Element of b
1 st
( b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
4 <> b
5 & b
4 <> b
6 & b
5 <> b
6 & b
2 "/\" b
3 = b
2 & b
2 "/\" b
4 = b
2 & b
2 "/\" b
5 = b
2 & b
3 "/\" b
6 = b
3 & b
4 "/\" b
6 = b
4 & b
5 "/\" b
6 = b
5 & b
3 "/\" b
4 = b
2 & b
3 "/\" b
5 = b
2 & b
4 "/\" b
5 = b
2 & b
3 "\/" b
4 = b
6 & b
3 "\/" b
5 = b
6 & b
4 "\/" b
5 = b
6 ) & ( for b
2 being
full Sublattice of b
1 holds
not
M_3 ,b
2 are_isomorphic ) ) )
:: deftheorem Def3 defines modular YELLOW11:def 3 :
Lemma13:
for b1 being LATTICE holds
( b1 is modular iff for b2, b3, b4, b5, b6 being Element of b1 holds
not ( b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b3 <> b4 & b3 <> b5 & b3 <> b6 & b4 <> b5 & b4 <> b6 & b5 <> b6 & b2 "/\" b3 = b2 & b2 "/\" b4 = b2 & b4 "/\" b6 = b4 & b5 "/\" b6 = b5 & b3 "/\" b4 = b2 & b3 "/\" b5 = b3 & b4 "/\" b5 = b2 & b3 "\/" b4 = b6 & b4 "\/" b5 = b6 ) )
theorem Th11: :: YELLOW11:11
Lemma14:
for b1 being LATTICE holds
( b1 is modular implies ( b1 is distributive iff for b2, b3, b4, b5, b6 being Element of b1 holds
not ( b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b3 <> b4 & b3 <> b5 & b3 <> b6 & b4 <> b5 & b4 <> b6 & b5 <> b6 & b2 "/\" b3 = b2 & b2 "/\" b4 = b2 & b2 "/\" b5 = b2 & b3 "/\" b6 = b3 & b4 "/\" b6 = b4 & b5 "/\" b6 = b5 & b3 "/\" b4 = b2 & b3 "/\" b5 = b2 & b4 "/\" b5 = b2 & b3 "\/" b4 = b6 & b3 "\/" b5 = b6 & b4 "\/" b5 = b6 ) ) )
theorem Th12: :: YELLOW11:12
:: deftheorem Def4 defines [# YELLOW11:def 4 :
:: deftheorem Def5 defines interval YELLOW11:def 5 :
theorem Th13: :: YELLOW11:13
theorem Th14: :: YELLOW11:14