:: BVFUNC24 semantic presentation
theorem Th1: :: BVFUNC24:1
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 implies
CompF b
3,b
2 = ((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b
9 )
theorem Th2: :: BVFUNC24:2
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 implies
CompF b
4,b
2 = ((((b3 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b
9 )
theorem Th3: :: BVFUNC24:3
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 implies
CompF b
5,b
2 = ((((b3 '/\' b4) '/\' b6) '/\' b7) '/\' b8) '/\' b
9 )
theorem Th4: :: BVFUNC24:4
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 implies
CompF b
6,b
2 = ((((b3 '/\' b4) '/\' b5) '/\' b7) '/\' b8) '/\' b
9 )
theorem Th5: :: BVFUNC24:5
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 implies
CompF b
7,b
2 = ((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b8) '/\' b
9 )
theorem Th6: :: BVFUNC24:6
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 implies
CompF b
8,b
2 = ((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b
9 )
theorem Th7: :: BVFUNC24:7
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 implies
CompF b
9,b
2 = ((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b
8 )
theorem Th8: :: BVFUNC24:8
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set for b
8 being
Functionfor b
9, b
10, b
11, b
12, b
13, b
14, b
15 being
set holds
( b
1 <> b
2 & b
1 <> b
3 & b
1 <> b
4 & b
1 <> b
5 & b
1 <> b
6 & b
1 <> b
7 & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
2 <> b
7 & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
5 <> b
6 & b
5 <> b
7 & b
6 <> b
7 & b
8 = ((((((b2 .--> b10) +* (b3 .--> b11)) +* (b4 .--> b12)) +* (b5 .--> b13)) +* (b6 .--> b14)) +* (b7 .--> b15)) +* (b1 .--> b9) implies ( b
8 . b
1 = b
9 & b
8 . b
2 = b
10 & b
8 . b
3 = b
11 & b
8 . b
4 = b
12 & b
8 . b
5 = b
13 & b
8 . b
6 = b
14 & b
8 . b
7 = b
15 ) )
theorem Th9: :: BVFUNC24:9
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set for b
8 being
Functionfor b
9, b
10, b
11, b
12, b
13, b
14, b
15 being
set holds
( b
8 = ((((((b2 .--> b10) +* (b3 .--> b11)) +* (b4 .--> b12)) +* (b5 .--> b13)) +* (b6 .--> b14)) +* (b7 .--> b15)) +* (b1 .--> b9) implies
dom b
8 = {b1,b2,b3,b4,b5,b6,b7} )
theorem Th10: :: BVFUNC24:10
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set for b
8 being
Functionfor b
9, b
10, b
11, b
12, b
13, b
14, b
15 being
set holds
( b
8 = ((((((b2 .--> b10) +* (b3 .--> b11)) +* (b4 .--> b12)) +* (b5 .--> b13)) +* (b6 .--> b14)) +* (b7 .--> b15)) +* (b1 .--> b9) implies
rng b
8 = {(b8 . b1),(b8 . b2),(b8 . b3),(b8 . b4),(b8 . b5),(b8 . b6),(b8 . b7)} )
theorem Th11: :: BVFUNC24:11
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
a_partition of b
1for b
10, b
11 being
Element of b
1 holds
not ( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8,b9} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 & not
EqClass b
11,
(((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) meets EqClass b
10,b
3 )
theorem Th12: :: BVFUNC24:12
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
a_partition of b
1for b
10, b
11 being
Element of b
1 holds
not ( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8,b9} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 &
EqClass b
10,
((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) = EqClass b
11,
((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) & not
EqClass b
11,
(CompF b3,b2) meets EqClass b
10,
(CompF b4,b2) )
theorem Th13: :: BVFUNC24:13
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
8 <> b
9 & b
8 <> b
10 & b
9 <> b
10 implies
CompF b
3,b
2 = (((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b
10 )
theorem Th14: :: BVFUNC24:14
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
8 <> b
9 & b
8 <> b
10 & b
9 <> b
10 implies
CompF b
4,b
2 = (((((b3 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b
10 )
theorem Th15: :: BVFUNC24:15
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
8 <> b
9 & b
8 <> b
10 & b
9 <> b
10 implies
CompF b
5,b
2 = (((((b3 '/\' b4) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b
10 )
theorem Th16: :: BVFUNC24:16
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
8 <> b
9 & b
8 <> b
10 & b
9 <> b
10 implies
CompF b
6,b
2 = (((((b3 '/\' b4) '/\' b5) '/\' b7) '/\' b8) '/\' b9) '/\' b
10 )
theorem Th17: :: BVFUNC24:17
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
8 <> b
9 & b
8 <> b
10 & b
9 <> b
10 implies
CompF b
7,b
2 = (((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b8) '/\' b9) '/\' b
10 )
theorem Th18: :: BVFUNC24:18
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
8 <> b
9 & b
8 <> b
10 & b
9 <> b
10 implies
CompF b
8,b
2 = (((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b9) '/\' b
10 )
theorem Th19: :: BVFUNC24:19
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
8 <> b
9 & b
8 <> b
10 & b
9 <> b
10 implies
CompF b
9,b
2 = (((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b
10 )
theorem Th20: :: BVFUNC24:20
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
8 <> b
9 & b
8 <> b
10 & b
9 <> b
10 implies
CompF b
10,b
2 = (((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b
9 )
theorem Th21: :: BVFUNC24:21
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set for b
9 being
Functionfor b
10, b
11, b
12, b
13, b
14, b
15, b
16, b
17 being
set holds
( b
1 <> b
2 & b
1 <> b
3 & b
1 <> b
4 & b
1 <> b
5 & b
1 <> b
6 & b
1 <> b
7 & b
1 <> b
8 & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
2 <> b
7 & b
2 <> b
8 & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
6 <> b
7 & b
6 <> b
8 & b
7 <> b
8 & b
9 = (((((((b2 .--> b11) +* (b3 .--> b12)) +* (b4 .--> b13)) +* (b5 .--> b14)) +* (b6 .--> b15)) +* (b7 .--> b16)) +* (b8 .--> b17)) +* (b1 .--> b10) implies ( b
9 . b
2 = b
11 & b
9 . b
3 = b
12 & b
9 . b
4 = b
13 & b
9 . b
5 = b
14 & b
9 . b
6 = b
15 & b
9 . b
7 = b
16 ) )
theorem Th22: :: BVFUNC24:22
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set for b
9 being
Functionfor b
10, b
11, b
12, b
13, b
14, b
15, b
16, b
17 being
set holds
( b
9 = (((((((b2 .--> b11) +* (b3 .--> b12)) +* (b4 .--> b13)) +* (b5 .--> b14)) +* (b6 .--> b15)) +* (b7 .--> b16)) +* (b8 .--> b17)) +* (b1 .--> b10) implies
dom b
9 = {b1,b2,b3,b4,b5,b6,b7,b8} )
theorem Th23: :: BVFUNC24:23
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set for b
9 being
Functionfor b
10, b
11, b
12, b
13, b
14, b
15, b
16, b
17 being
set holds
( b
9 = (((((((b2 .--> b11) +* (b3 .--> b12)) +* (b4 .--> b13)) +* (b5 .--> b14)) +* (b6 .--> b15)) +* (b7 .--> b16)) +* (b8 .--> b17)) +* (b1 .--> b10) implies
rng b
9 = {(b9 . b1),(b9 . b2),(b9 . b3),(b9 . b4),(b9 . b5),(b9 . b6),(b9 . b7),(b9 . b8)} )
theorem Th24: :: BVFUNC24:24
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
a_partition of b
1for b
11, b
12 being
Element of b
1 holds
not ( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8,b9,b10} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
8 <> b
9 & b
8 <> b
10 & b
9 <> b
10 & not
(EqClass b12,((((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10)) /\ (EqClass b11,b3) <> {} )
theorem Th25: :: BVFUNC24:25
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
a_partition of b
1for b
11, b
12 being
Element of b
1 holds
not ( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8,b9,b10} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
8 <> b
9 & b
8 <> b
10 & b
9 <> b
10 &
EqClass b
11,
(((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) = EqClass b
12,
(((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) & not
EqClass b
12,
(CompF b3,b2) meets EqClass b
11,
(CompF b4,b2) )
definition
canceled;
end;
:: deftheorem Def1 BVFUNC24:def 1 :
canceled;
Lemma20:
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set holds {b1,b2,b3,b4,b5,b6,b7,b8,b9} = {b1,b2,b3,b4} \/ {b5,b6,b7,b8,b9}
theorem Th26: :: BVFUNC24:26
canceled;
theorem Th27: :: BVFUNC24:27
canceled;
theorem Th28: :: BVFUNC24:28
canceled;
theorem Th29: :: BVFUNC24:29
canceled;
theorem Th30: :: BVFUNC24:30
canceled;
theorem Th31: :: BVFUNC24:31
canceled;
theorem Th32: :: BVFUNC24:32
canceled;
theorem Th33: :: BVFUNC24:33
canceled;
theorem Th34: :: BVFUNC24:34
canceled;
theorem Th35: :: BVFUNC24:35
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
3 <> b
11 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
4 <> b
11 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
5 <> b
11 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
6 <> b
11 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
7 <> b
11 & b
8 <> b
9 & b
8 <> b
10 & b
8 <> b
11 & b
9 <> b
10 & b
9 <> b
11 & b
10 <> b
11 implies
CompF b
3,b
2 = ((((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b
11 )
theorem Th36: :: BVFUNC24:36
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
3 <> b
11 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
4 <> b
11 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
5 <> b
11 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
6 <> b
11 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
7 <> b
11 & b
8 <> b
9 & b
8 <> b
10 & b
8 <> b
11 & b
9 <> b
10 & b
9 <> b
11 & b
10 <> b
11 implies
CompF b
4,b
2 = ((((((b3 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b
11 )
theorem Th37: :: BVFUNC24:37
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
3 <> b
11 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
4 <> b
11 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
5 <> b
11 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
6 <> b
11 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
7 <> b
11 & b
8 <> b
9 & b
8 <> b
10 & b
8 <> b
11 & b
9 <> b
10 & b
9 <> b
11 & b
10 <> b
11 implies
CompF b
5,b
2 = ((((((b3 '/\' b4) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b
11 )
theorem Th38: :: BVFUNC24:38
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
3 <> b
11 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
4 <> b
11 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
5 <> b
11 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
6 <> b
11 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
7 <> b
11 & b
8 <> b
9 & b
8 <> b
10 & b
8 <> b
11 & b
9 <> b
10 & b
9 <> b
11 & b
10 <> b
11 implies
CompF b
6,b
2 = ((((((b3 '/\' b4) '/\' b5) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b
11 )
theorem Th39: :: BVFUNC24:39
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
3 <> b
11 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
4 <> b
11 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
5 <> b
11 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
6 <> b
11 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
7 <> b
11 & b
8 <> b
9 & b
8 <> b
10 & b
8 <> b
11 & b
9 <> b
10 & b
9 <> b
11 & b
10 <> b
11 implies
CompF b
7,b
2 = ((((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b8) '/\' b9) '/\' b10) '/\' b
11 )
theorem Th40: :: BVFUNC24:40
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
3 <> b
11 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
4 <> b
11 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
5 <> b
11 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
6 <> b
11 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
7 <> b
11 & b
8 <> b
9 & b
8 <> b
10 & b
8 <> b
11 & b
9 <> b
10 & b
9 <> b
11 & b
10 <> b
11 implies
CompF b
8,b
2 = ((((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b9) '/\' b10) '/\' b
11 )
theorem Th41: :: BVFUNC24:41
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
3 <> b
11 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
4 <> b
11 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
5 <> b
11 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
6 <> b
11 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
7 <> b
11 & b
8 <> b
9 & b
8 <> b
10 & b
8 <> b
11 & b
9 <> b
10 & b
9 <> b
11 & b
10 <> b
11 implies
CompF b
9,b
2 = ((((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b10) '/\' b
11 )
theorem Th42: :: BVFUNC24:42
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
3 <> b
11 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
4 <> b
11 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
5 <> b
11 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
6 <> b
11 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
7 <> b
11 & b
8 <> b
9 & b
8 <> b
10 & b
8 <> b
11 & b
9 <> b
10 & b
9 <> b
11 & b
10 <> b
11 implies
CompF b
10,b
2 = ((((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b
11 )
theorem Th43: :: BVFUNC24:43
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
a_partition of b
1 holds
( b
2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
3 <> b
11 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
4 <> b
11 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
5 <> b
11 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
6 <> b
11 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
7 <> b
11 & b
8 <> b
9 & b
8 <> b
10 & b
8 <> b
11 & b
9 <> b
10 & b
9 <> b
11 & b
10 <> b
11 implies
CompF b
11,b
2 = ((((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b
10 )
theorem Th44: :: BVFUNC24:44
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set for b
10 being
Functionfor b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19 being
set holds
( b
1 <> b
2 & b
1 <> b
3 & b
1 <> b
4 & b
1 <> b
5 & b
1 <> b
6 & b
1 <> b
7 & b
1 <> b
8 & b
1 <> b
9 & b
2 <> b
3 & b
2 <> b
4 & b
2 <> b
5 & b
2 <> b
6 & b
2 <> b
7 & b
2 <> b
8 & b
2 <> b
9 & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
7 <> b
8 & b
7 <> b
9 & b
8 <> b
9 & b
10 = ((((((((b2 .--> b12) +* (b3 .--> b13)) +* (b4 .--> b14)) +* (b5 .--> b15)) +* (b6 .--> b16)) +* (b7 .--> b17)) +* (b8 .--> b18)) +* (b9 .--> b19)) +* (b1 .--> b11) implies ( b
10 . b
1 = b
11 & b
10 . b
2 = b
12 & b
10 . b
3 = b
13 & b
10 . b
4 = b
14 & b
10 . b
5 = b
15 & b
10 . b
6 = b
16 & b
10 . b
7 = b
17 & b
10 . b
8 = b
18 & b
10 . b
9 = b
19 ) )
theorem Th45: :: BVFUNC24:45
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set for b
10 being
Functionfor b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19 being
set holds
( b
10 = ((((((((b2 .--> b12) +* (b3 .--> b13)) +* (b4 .--> b14)) +* (b5 .--> b15)) +* (b6 .--> b16)) +* (b7 .--> b17)) +* (b8 .--> b18)) +* (b9 .--> b19)) +* (b1 .--> b11) implies
dom b
10 = {b1,b2,b3,b4,b5,b6,b7,b8,b9} )
theorem Th46: :: BVFUNC24:46
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set for b
10 being
Functionfor b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19 being
set holds
( b
10 = ((((((((b2 .--> b12) +* (b3 .--> b13)) +* (b4 .--> b14)) +* (b5 .--> b15)) +* (b6 .--> b16)) +* (b7 .--> b17)) +* (b8 .--> b18)) +* (b9 .--> b19)) +* (b1 .--> b11) implies
rng b
10 = {(b10 . b1),(b10 . b2),(b10 . b3),(b10 . b4),(b10 . b5),(b10 . b6),(b10 . b7),(b10 . b8),(b10 . b9)} )
theorem Th47: :: BVFUNC24:47
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
a_partition of b
1for b
12, b
13 being
Element of b
1 holds
not ( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
3 <> b
11 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
4 <> b
11 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
5 <> b
11 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
6 <> b
11 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
7 <> b
11 & b
8 <> b
9 & b
8 <> b
10 & b
8 <> b
11 & b
9 <> b
10 & b
9 <> b
11 & b
10 <> b
11 & not
(EqClass b13,(((((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b11)) /\ (EqClass b12,b3) <> {} )
theorem Th48: :: BVFUNC24:48
for b
1 being non
empty set for b
2 being
Subset of
(PARTITIONS b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
a_partition of b
1for b
12, b
13 being
Element of b
1 holds
not ( b
2 is
independent & b
2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} & b
3 <> b
4 & b
3 <> b
5 & b
3 <> b
6 & b
3 <> b
7 & b
3 <> b
8 & b
3 <> b
9 & b
3 <> b
10 & b
3 <> b
11 & b
4 <> b
5 & b
4 <> b
6 & b
4 <> b
7 & b
4 <> b
8 & b
4 <> b
9 & b
4 <> b
10 & b
4 <> b
11 & b
5 <> b
6 & b
5 <> b
7 & b
5 <> b
8 & b
5 <> b
9 & b
5 <> b
10 & b
5 <> b
11 & b
6 <> b
7 & b
6 <> b
8 & b
6 <> b
9 & b
6 <> b
10 & b
6 <> b
11 & b
7 <> b
8 & b
7 <> b
9 & b
7 <> b
10 & b
7 <> b
11 & b
8 <> b
9 & b
8 <> b
10 & b
8 <> b
11 & b
9 <> b
10 & b
9 <> b
11 & b
10 <> b
11 &
EqClass b
12,
((((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b11) = EqClass b
13,
((((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b11) & not
EqClass b
13,
(CompF b3,b2) meets EqClass b
12,
(CompF b4,b2) )