:: WAYBEL35 semantic presentation
Lemma1:
for b1, b2, b3 being set holds
not ( b2 in {b1} \/ b3 & not b2 = b1 & not b2 in b3 )
Lemma2:
for b1 being trivial set
for b2 being Subset of b1 holds b2 is trivial
theorem Th1: :: WAYBEL35:1
theorem Th2: :: WAYBEL35:2
theorem Th3: :: WAYBEL35:3
:: deftheorem Def1 defines extra-order WAYBEL35:def 1 :
:: deftheorem Def2 defines -LowerMap WAYBEL35:def 2 :
:: deftheorem Def3 defines strict_chain WAYBEL35:def 3 :
theorem Th4: :: WAYBEL35:4
theorem Th5: :: WAYBEL35:5
theorem Th6: :: WAYBEL35:6
theorem Th7: :: WAYBEL35:7
theorem Th8: :: WAYBEL35:8
:: deftheorem Def4 defines maximal WAYBEL35:def 4 :
:: deftheorem Def5 defines Strict_Chains WAYBEL35:def 5 :
theorem Th9: :: WAYBEL35:9
theorem Th10: :: WAYBEL35:10
Lemma15:
for b1 being non empty Poset
for b2 being auxiliary(i) auxiliary(ii) Relation of b1
for b3 being non empty strict_chain of b2
for b4 being Subset of b3 holds
not ( ex_sup_of b4,b1 & b3 is maximal & not "\/" b4,b1 in b3 & ( for b5 being Element of b1 holds
not ( b5 in b3 & "\/" b4,b1 < b5 & not [("\/" b4,b1),b5] in b2 & ex b6 being Element of (subrelstr b3) st
( b5 = b6 & b4 is_<=_than b6 & ( for b7 being Element of (subrelstr b3) holds
( b4 is_<=_than b7 implies b6 <= b7 ) ) ) ) ) )
theorem Th11: :: WAYBEL35:11
theorem Th12: :: WAYBEL35:12
for b
1 being non
empty Posetfor b
2 being
auxiliary(i) auxiliary(ii) Relation of b
1for b
3 being non
empty strict_chain of b
2for b
4 being
Subset of b
3 holds
(
ex_inf_of (uparrow ("\/" b4,b1)) /\ b
3,b
1 &
ex_sup_of b
4,b
1 & b
3 is
maximal implies (
"\/" b
4,
(subrelstr b3) = "/\" ((uparrow ("\/" b4,b1)) /\ b3),b
1 & ( not
"\/" b
4,b
1 in b
3 implies
"\/" b
4,b
1 < "/\" ((uparrow ("\/" b4,b1)) /\ b3),b
1 ) ) )
theorem Th13: :: WAYBEL35:13
theorem Th14: :: WAYBEL35:14
theorem Th15: :: WAYBEL35:15
:: deftheorem Def6 defines satisfies_SIC_on WAYBEL35:def 6 :
:: deftheorem Def7 defines satisfying_SIC WAYBEL35:def 7 :
theorem Th16: :: WAYBEL35:16
theorem Th17: :: WAYBEL35:17
:: deftheorem Def8 defines SetBelow WAYBEL35:def 8 :
theorem Th18: :: WAYBEL35:18
theorem Th19: :: WAYBEL35:19
theorem Th20: :: WAYBEL35:20
theorem Th21: :: WAYBEL35:21
:: deftheorem Def9 defines sup-closed WAYBEL35:def 9 :
theorem Th22: :: WAYBEL35:22
theorem Th23: :: WAYBEL35:23
theorem Th24: :: WAYBEL35:24
definition
let c
1 be non
empty RelStr ;
let c
2 be
Relation of the
carrier of c
1;
let c
3 be
set ;
func SupBelow c
2,c
3 -> set means :
Def10:
:: WAYBEL35:def 10
for b
1 being
set holds
( b
1 in a
4 iff b
1 = sup (SetBelow a2,a3,b1) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff b2 = sup (SetBelow c2,c3,b2) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff b3 = sup (SetBelow c2,c3,b3) ) ) & ( for b3 being set holds
( b3 in b2 iff b3 = sup (SetBelow c2,c3,b3) ) ) implies b1 = b2 )
end;
:: deftheorem Def10 defines SupBelow WAYBEL35:def 10 :
theorem Th25: :: WAYBEL35:25
theorem Th26: :: WAYBEL35:26
theorem Th27: :: WAYBEL35:27
theorem Th28: :: WAYBEL35:28
theorem Th29: :: WAYBEL35:29