:: MSSUBFAM semantic presentation
scheme :: MSSUBFAM:sch 1
s1{ F
1()
-> set , F
2()
-> ManySortedSet of F
1(), F
3()
-> ManySortedSet of F
1(), P
1[
set ,
set ,
set ] } :
ex b
1 being
ManySortedFunction of F
2(),F
3() st
for b
2 being
set holds
not ( b
2 in F
1() & ( for b
3 being
Function of F
2()
. b
2,F
3()
. b
2 holds
not ( b
3 = b
1 . b
2 & ( for b
4 being
set holds
( b
4 in F
2()
. b
2 implies P
1[b
3 . b
4,b
4,b
2] ) ) ) ) )
provided
E1:
for b
1 being
set holds
( b
1 in F
1() implies for b
2 being
set holds
not ( b
2 in F
2()
. b
1 & ( for b
3 being
set holds
not ( b
3 in F
3()
. b
1 & P
1[b
3,b
2,b
1] ) ) ) )
theorem Th1: :: MSSUBFAM:1
theorem Th2: :: MSSUBFAM:2
theorem Th3: :: MSSUBFAM:3
theorem Th4: :: MSSUBFAM:4
theorem Th5: :: MSSUBFAM:5
theorem Th6: :: MSSUBFAM:6
theorem Th7: :: MSSUBFAM:7
theorem Th8: :: MSSUBFAM:8
theorem Th9: :: MSSUBFAM:9
theorem Th10: :: MSSUBFAM:10
theorem Th11: :: MSSUBFAM:11
theorem Th12: :: MSSUBFAM:12
theorem Th13: :: MSSUBFAM:13
theorem Th14: :: MSSUBFAM:14
theorem Th15: :: MSSUBFAM:15
theorem Th16: :: MSSUBFAM:16
theorem Th17: :: MSSUBFAM:17
theorem Th18: :: MSSUBFAM:18
theorem Th19: :: MSSUBFAM:19
theorem Th20: :: MSSUBFAM:20
theorem Th21: :: MSSUBFAM:21
theorem Th22: :: MSSUBFAM:22
theorem Th23: :: MSSUBFAM:23
theorem Th24: :: MSSUBFAM:24
theorem Th25: :: MSSUBFAM:25
theorem Th26: :: MSSUBFAM:26
theorem Th27: :: MSSUBFAM:27
theorem Th28: :: MSSUBFAM:28
theorem Th29: :: MSSUBFAM:29
theorem Th30: :: MSSUBFAM:30
theorem Th31: :: MSSUBFAM:31
theorem Th32: :: MSSUBFAM:32
theorem Th33: :: MSSUBFAM:33
theorem Th34: :: MSSUBFAM:34
theorem Th35: :: MSSUBFAM:35
theorem Th36: :: MSSUBFAM:36
theorem Th37: :: MSSUBFAM:37
theorem Th38: :: MSSUBFAM:38
theorem Th39: :: MSSUBFAM:39
theorem Th40: :: MSSUBFAM:40
:: deftheorem Def1 MSSUBFAM:def 1 :
canceled;
:: deftheorem Def2 defines meet MSSUBFAM:def 2 :
theorem Th41: :: MSSUBFAM:41
theorem Th42: :: MSSUBFAM:42
theorem Th43: :: MSSUBFAM:43
theorem Th44: :: MSSUBFAM:44
theorem Th45: :: MSSUBFAM:45
theorem Th46: :: MSSUBFAM:46
theorem Th47: :: MSSUBFAM:47
theorem Th48: :: MSSUBFAM:48
theorem Th49: :: MSSUBFAM:49
theorem Th50: :: MSSUBFAM:50
theorem Th51: :: MSSUBFAM:51
theorem Th52: :: MSSUBFAM:52
theorem Th53: :: MSSUBFAM:53
:: deftheorem Def3 defines additive MSSUBFAM:def 3 :
:: deftheorem Def4 defines absolutely-additive MSSUBFAM:def 4 :
:: deftheorem Def5 defines multiplicative MSSUBFAM:def 5 :
:: deftheorem Def6 defines absolutely-multiplicative MSSUBFAM:def 6 :
:: deftheorem Def7 defines properly-upper-bound MSSUBFAM:def 7 :
:: deftheorem Def8 defines properly-lower-bound MSSUBFAM:def 8 :
Lemma19:
for b1 being set
for b2 being ManySortedSet of b1 holds
( bool b2 is additive & bool b2 is absolutely-additive & bool b2 is multiplicative & bool b2 is absolutely-multiplicative & bool b2 is properly-upper-bound & bool b2 is properly-lower-bound )