:: BVFUNC13 semantic presentation
theorem Th1: :: BVFUNC13:1
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All ('not' (All b2,b4,b3)),b
5,b
3 '<' 'not' (All (All b2,b5,b3),b4,b3) )
theorem Th2: :: BVFUNC13:2
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All (All ('not' b2),b4,b3),b
5,b
3 '<' 'not' (All (All b2,b5,b3),b4,b3)
theorem Th3: :: BVFUNC13:3
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All ('not' (Ex b2,b4,b3)),b
5,b
3 '<' 'not' (All (All b2,b5,b3),b4,b3)
theorem Th4: :: BVFUNC13:4
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All (Ex ('not' b2),b4,b3),b
5,b
3 '<' 'not' (All (All b2,b5,b3),b4,b3) )
theorem Th5: :: BVFUNC13:5
canceled;
theorem Th6: :: BVFUNC13:6
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex (All ('not' b2),b4,b3),b
5,b
3 '<' 'not' (All (All b2,b5,b3),b4,b3) )
theorem Th7: :: BVFUNC13:7
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex ('not' (Ex b2,b4,b3)),b
5,b
3 '<' 'not' (All (All b2,b5,b3),b4,b3) )
theorem Th8: :: BVFUNC13:8
canceled;
theorem Th9: :: BVFUNC13:9
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (All (Ex b2,b4,b3),b5,b3) '<' 'not' (Ex (All b2,b5,b3),b4,b3) )
theorem Th10: :: BVFUNC13:10
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (Ex (Ex b2,b4,b3),b5,b3) '<' 'not' (Ex (All b2,b5,b3),b4,b3) )
theorem Th11: :: BVFUNC13:11
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
'not' (Ex (Ex b2,b4,b3),b5,b3) '<' 'not' (All (Ex b2,b5,b3),b4,b3)
theorem Th12: :: BVFUNC13:12
canceled;
theorem Th13: :: BVFUNC13:13
canceled;
theorem Th14: :: BVFUNC13:14
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (Ex (All b2,b4,b3),b5,b3) '<' 'not' (All (All b2,b5,b3),b4,b3) )
theorem Th15: :: BVFUNC13:15
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (All (Ex b2,b4,b3),b5,b3) '<' 'not' (All (All b2,b5,b3),b4,b3) )
theorem Th16: :: BVFUNC13:16
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
'not' (Ex (Ex b2,b4,b3),b5,b3) '<' 'not' (All (All b2,b5,b3),b4,b3)
theorem Th17: :: BVFUNC13:17
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (Ex (All b2,b4,b3),b5,b3) '<' Ex ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th18: :: BVFUNC13:18
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (All (Ex b2,b4,b3),b5,b3) '<' Ex ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th19: :: BVFUNC13:19
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
'not' (Ex (Ex b2,b4,b3),b5,b3) '<' Ex ('not' (All b2,b5,b3)),b
4,b
3
theorem Th20: :: BVFUNC13:20
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (All (Ex b2,b4,b3),b5,b3) '<' All ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th21: :: BVFUNC13:21
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (Ex (Ex b2,b4,b3),b5,b3) '<' All ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th22: :: BVFUNC13:22
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
'not' (Ex (Ex b2,b4,b3),b5,b3) '<' Ex ('not' (Ex b2,b5,b3)),b
4,b
3
theorem Th23: :: BVFUNC13:23
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (Ex (Ex b2,b4,b3),b5,b3) = All ('not' (Ex b2,b5,b3)),b
4,b
3 )
theorem Th24: :: BVFUNC13:24
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (All (Ex b2,b4,b3),b5,b3) '<' Ex (Ex ('not' b2),b5,b3),b
4,b
3 )
theorem Th25: :: BVFUNC13:25
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
'not' (Ex (Ex b2,b4,b3),b5,b3) '<' Ex (Ex ('not' b2),b5,b3),b
4,b
3
theorem Th26: :: BVFUNC13:26
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (All (Ex b2,b4,b3),b5,b3) '<' All (Ex ('not' b2),b5,b3),b
4,b
3 )
theorem Th27: :: BVFUNC13:27
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (Ex (Ex b2,b4,b3),b5,b3) '<' All (Ex ('not' b2),b5,b3),b
4,b
3 )
theorem Th28: :: BVFUNC13:28
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
'not' (Ex (Ex b2,b4,b3),b5,b3) '<' Ex (All ('not' b2),b5,b3),b
4,b
3
theorem Th29: :: BVFUNC13:29
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
'not' (Ex (Ex b2,b4,b3),b5,b3) = All (All ('not' b2),b5,b3),b
4,b
3 )
theorem Th30: :: BVFUNC13:30
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex ('not' (Ex b2,b4,b3)),b
5,b
3 '<' 'not' (Ex (All b2,b5,b3),b4,b3) )
theorem Th31: :: BVFUNC13:31
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All ('not' (Ex b2,b4,b3)),b
5,b
3 '<' 'not' (Ex (All b2,b5,b3),b4,b3)
theorem Th32: :: BVFUNC13:32
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All ('not' (Ex b2,b4,b3)),b
5,b
3 '<' 'not' (All (Ex b2,b5,b3),b4,b3)
theorem Th33: :: BVFUNC13:33
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All ('not' (Ex b2,b4,b3)),b
5,b
3 = 'not' (Ex (Ex b2,b5,b3),b4,b3) )
theorem Th34: :: BVFUNC13:34
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex ('not' (All b2,b4,b3)),b
5,b
3 = Ex ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th35: :: BVFUNC13:35
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All ('not' (All b2,b4,b3)),b
5,b
3 '<' Ex ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th36: :: BVFUNC13:36
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex ('not' (Ex b2,b4,b3)),b
5,b
3 '<' Ex ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th37: :: BVFUNC13:37
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All ('not' (Ex b2,b4,b3)),b
5,b
3 '<' Ex ('not' (All b2,b5,b3)),b
4,b
3
theorem Th38: :: BVFUNC13:38
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex ('not' (Ex b2,b4,b3)),b
5,b
3 '<' All ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th39: :: BVFUNC13:39
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All ('not' (Ex b2,b4,b3)),b
5,b
3 '<' All ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th40: :: BVFUNC13:40
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All ('not' (Ex b2,b4,b3)),b
5,b
3 '<' Ex ('not' (Ex b2,b5,b3)),b
4,b
3
theorem Th41: :: BVFUNC13:41
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All ('not' (Ex b2,b4,b3)),b
5,b
3 = All ('not' (Ex b2,b5,b3)),b
4,b
3 )
theorem Th42: :: BVFUNC13:42
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex ('not' (Ex b2,b4,b3)),b
5,b
3 '<' Ex (Ex ('not' b2),b5,b3),b
4,b
3 )
theorem Th43: :: BVFUNC13:43
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All ('not' (Ex b2,b4,b3)),b
5,b
3 '<' Ex (Ex ('not' b2),b5,b3),b
4,b
3
theorem Th44: :: BVFUNC13:44
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex ('not' (Ex b2,b4,b3)),b
5,b
3 '<' All (Ex ('not' b2),b5,b3),b
4,b
3 )
theorem Th45: :: BVFUNC13:45
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All ('not' (Ex b2,b4,b3)),b
5,b
3 '<' All (Ex ('not' b2),b5,b3),b
4,b
3 )
theorem Th46: :: BVFUNC13:46
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All ('not' (Ex b2,b4,b3)),b
5,b
3 '<' Ex (All ('not' b2),b5,b3),b
4,b
3
theorem Th47: :: BVFUNC13:47
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All ('not' (Ex b2,b4,b3)),b
5,b
3 = All (All ('not' b2),b5,b3),b
4,b
3 )
theorem Th48: :: BVFUNC13:48
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex (All ('not' b2),b4,b3),b
5,b
3 '<' 'not' (Ex (All b2,b5,b3),b4,b3) )
theorem Th49: :: BVFUNC13:49
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All (All ('not' b2),b4,b3),b
5,b
3 '<' 'not' (Ex (All b2,b5,b3),b4,b3) )
theorem Th50: :: BVFUNC13:50
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All (All ('not' b2),b4,b3),b
5,b
3 '<' 'not' (All (Ex b2,b5,b3),b4,b3)
theorem Th51: :: BVFUNC13:51
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All (All ('not' b2),b4,b3),b
5,b
3 '<' 'not' (Ex (Ex b2,b5,b3),b4,b3) )
theorem Th52: :: BVFUNC13:52
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex (Ex ('not' b2),b4,b3),b
5,b
3 '<' Ex ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th53: :: BVFUNC13:53
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All (Ex ('not' b2),b4,b3),b
5,b
3 '<' Ex ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th54: :: BVFUNC13:54
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex (All ('not' b2),b4,b3),b
5,b
3 '<' Ex ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th55: :: BVFUNC13:55
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All (All ('not' b2),b4,b3),b
5,b
3 '<' Ex ('not' (All b2,b5,b3)),b
4,b
3
theorem Th56: :: BVFUNC13:56
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex (All ('not' b2),b4,b3),b
5,b
3 '<' All ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th57: :: BVFUNC13:57
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All (All ('not' b2),b4,b3),b
5,b
3 '<' All ('not' (All b2,b5,b3)),b
4,b
3 )
theorem Th58: :: BVFUNC13:58
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All (All ('not' b2),b4,b3),b
5,b
3 '<' Ex ('not' (Ex b2,b5,b3)),b
4,b
3
theorem Th59: :: BVFUNC13:59
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
All (All ('not' b2),b4,b3),b
5,b
3 = All ('not' (Ex b2,b5,b3)),b
4,b
3 )
theorem Th60: :: BVFUNC13:60
canceled;
theorem Th61: :: BVFUNC13:61
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
All (Ex ('not' b2),b4,b3),b
5,b
3 '<' Ex (Ex ('not' b2),b5,b3),b
4,b
3
theorem Th62: :: BVFUNC13:62
for b
1 being non
empty set for b
2 being
Element of
Funcs b
1,
BOOLEAN for b
3 being
Subset of
(PARTITIONS b1)for b
4, b
5 being
a_partition of b
1 holds
( b
3 is
independent implies
Ex (All ('not' b2),b4,b3),b
5,b
3 '<' Ex (Ex ('not' b2),b5,b3),b
4,b
3 )