:: BVFUNC24 semantic presentation
theorem Th1: :: BVFUNC24:1
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 holds
CompF b3,
b2 = ((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9
theorem Th2: :: BVFUNC24:2
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 holds
CompF b4,
b2 = ((((b3 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9
theorem Th3: :: BVFUNC24:3
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 holds
CompF b5,
b2 = ((((b3 '/\' b4) '/\' b6) '/\' b7) '/\' b8) '/\' b9
theorem Th4: :: BVFUNC24:4
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 holds
CompF b6,
b2 = ((((b3 '/\' b4) '/\' b5) '/\' b7) '/\' b8) '/\' b9
theorem Th5: :: BVFUNC24:5
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 holds
CompF b7,
b2 = ((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b8) '/\' b9
theorem Th6: :: BVFUNC24:6
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 holds
CompF b8,
b2 = ((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b9
theorem Th7: :: BVFUNC24:7
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 holds
CompF b9,
b2 = ((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b8
theorem Th8: :: BVFUNC24:8
for
b1,
b2,
b3,
b4,
b5,
b6,
b7 being
set for
b8 being
Functionfor
b9,
b10,
b11,
b12,
b13,
b14,
b15 being
set st
b1 <> b2 &
b1 <> b3 &
b1 <> b4 &
b1 <> b5 &
b1 <> b6 &
b1 <> b7 &
b2 <> b3 &
b2 <> b4 &
b2 <> b5 &
b2 <> b6 &
b2 <> b7 &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b5 <> b6 &
b5 <> b7 &
b6 <> b7 &
b8 = ((((((b2 .--> b10) +* (b3 .--> b11)) +* (b4 .--> b12)) +* (b5 .--> b13)) +* (b6 .--> b14)) +* (b7 .--> b15)) +* (b1 .--> b9) holds
(
b8 . b1 = b9 &
b8 . b2 = b10 &
b8 . b3 = b11 &
b8 . b4 = b12 &
b8 . b5 = b13 &
b8 . b6 = b14 &
b8 . b7 = b15 )
theorem Th9: :: BVFUNC24:9
for
b1,
b2,
b3,
b4,
b5,
b6,
b7 being
set for
b8 being
Functionfor
b9,
b10,
b11,
b12,
b13,
b14,
b15 being
set st
b8 = ((((((b2 .--> b10) +* (b3 .--> b11)) +* (b4 .--> b12)) +* (b5 .--> b13)) +* (b6 .--> b14)) +* (b7 .--> b15)) +* (b1 .--> b9) holds
dom b8 = {b1,b2,b3,b4,b5,b6,b7}
theorem Th10: :: BVFUNC24:10
for
b1,
b2,
b3,
b4,
b5,
b6,
b7 being
set for
b8 being
Functionfor
b9,
b10,
b11,
b12,
b13,
b14,
b15 being
set st
b8 = ((((((b2 .--> b10) +* (b3 .--> b11)) +* (b4 .--> b12)) +* (b5 .--> b13)) +* (b6 .--> b14)) +* (b7 .--> b15)) +* (b1 .--> b9) holds
rng b8 = {(b8 . b1),(b8 . b2),(b8 . b3),(b8 . b4),(b8 . b5),(b8 . b6),(b8 . b7)}
theorem Th11: :: BVFUNC24:11
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
a_partition of
b1for
b10,
b11 being
Element of
b1 st
b2 is
independent &
b2 = {b3,b4,b5,b6,b7,b8,b9} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 holds
EqClass b11,
(((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) meets EqClass b10,
b3
theorem Th12: :: BVFUNC24:12
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
a_partition of
b1for
b10,
b11 being
Element of
b1 st
b2 is
independent &
b2 = {b3,b4,b5,b6,b7,b8,b9} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 &
EqClass b10,
((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) = EqClass b11,
((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) holds
EqClass b11,
(CompF b3,b2) meets EqClass b10,
(CompF b4,b2)
theorem Th13: :: BVFUNC24:13
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b8 <> b9 &
b8 <> b10 &
b9 <> b10 holds
CompF b3,
b2 = (((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10
theorem Th14: :: BVFUNC24:14
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b8 <> b9 &
b8 <> b10 &
b9 <> b10 holds
CompF b4,
b2 = (((((b3 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10
theorem Th15: :: BVFUNC24:15
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b8 <> b9 &
b8 <> b10 &
b9 <> b10 holds
CompF b5,
b2 = (((((b3 '/\' b4) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10
theorem Th16: :: BVFUNC24:16
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b8 <> b9 &
b8 <> b10 &
b9 <> b10 holds
CompF b6,
b2 = (((((b3 '/\' b4) '/\' b5) '/\' b7) '/\' b8) '/\' b9) '/\' b10
theorem Th17: :: BVFUNC24:17
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b8 <> b9 &
b8 <> b10 &
b9 <> b10 holds
CompF b7,
b2 = (((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b8) '/\' b9) '/\' b10
theorem Th18: :: BVFUNC24:18
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b8 <> b9 &
b8 <> b10 &
b9 <> b10 holds
CompF b8,
b2 = (((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b9) '/\' b10
theorem Th19: :: BVFUNC24:19
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b8 <> b9 &
b8 <> b10 &
b9 <> b10 holds
CompF b9,
b2 = (((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b10
theorem Th20: :: BVFUNC24:20
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b8 <> b9 &
b8 <> b10 &
b9 <> b10 holds
CompF b10,
b2 = (((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9
theorem Th21: :: BVFUNC24:21
for
b1,
b2,
b3,
b4,
b5,
b6,
b7,
b8 being
set for
b9 being
Functionfor
b10,
b11,
b12,
b13,
b14,
b15,
b16,
b17 being
set st
b1 <> b2 &
b1 <> b3 &
b1 <> b4 &
b1 <> b5 &
b1 <> b6 &
b1 <> b7 &
b1 <> b8 &
b2 <> b3 &
b2 <> b4 &
b2 <> b5 &
b2 <> b6 &
b2 <> b7 &
b2 <> b8 &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b6 <> b7 &
b6 <> b8 &
b7 <> b8 &
b9 = (((((((b2 .--> b11) +* (b3 .--> b12)) +* (b4 .--> b13)) +* (b5 .--> b14)) +* (b6 .--> b15)) +* (b7 .--> b16)) +* (b8 .--> b17)) +* (b1 .--> b10) holds
(
b9 . b2 = b11 &
b9 . b3 = b12 &
b9 . b4 = b13 &
b9 . b5 = b14 &
b9 . b6 = b15 &
b9 . b7 = b16 )
theorem Th22: :: BVFUNC24:22
for
b1,
b2,
b3,
b4,
b5,
b6,
b7,
b8 being
set for
b9 being
Functionfor
b10,
b11,
b12,
b13,
b14,
b15,
b16,
b17 being
set st
b9 = (((((((b2 .--> b11) +* (b3 .--> b12)) +* (b4 .--> b13)) +* (b5 .--> b14)) +* (b6 .--> b15)) +* (b7 .--> b16)) +* (b8 .--> b17)) +* (b1 .--> b10) holds
dom b9 = {b1,b2,b3,b4,b5,b6,b7,b8}
theorem Th23: :: BVFUNC24:23
for
b1,
b2,
b3,
b4,
b5,
b6,
b7,
b8 being
set for
b9 being
Functionfor
b10,
b11,
b12,
b13,
b14,
b15,
b16,
b17 being
set st
b9 = (((((((b2 .--> b11) +* (b3 .--> b12)) +* (b4 .--> b13)) +* (b5 .--> b14)) +* (b6 .--> b15)) +* (b7 .--> b16)) +* (b8 .--> b17)) +* (b1 .--> b10) holds
rng b9 = {(b9 . b1),(b9 . b2),(b9 . b3),(b9 . b4),(b9 . b5),(b9 . b6),(b9 . b7),(b9 . b8)}
theorem Th24: :: BVFUNC24:24
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10 being
a_partition of
b1for
b11,
b12 being
Element of
b1 st
b2 is
independent &
b2 = {b3,b4,b5,b6,b7,b8,b9,b10} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b8 <> b9 &
b8 <> b10 &
b9 <> b10 holds
(EqClass b12,((((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10)) /\ (EqClass b11,b3) <> {}
theorem Th25: :: BVFUNC24:25
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10 being
a_partition of
b1for
b11,
b12 being
Element of
b1 st
b2 is
independent &
b2 = {b3,b4,b5,b6,b7,b8,b9,b10} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b8 <> b9 &
b8 <> b10 &
b9 <> b10 &
EqClass b11,
(((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) = EqClass b12,
(((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) holds
EqClass b12,
(CompF b3,b2) meets EqClass b11,
(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
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10,
b11 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b3 <> b11 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b4 <> b11 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b5 <> b11 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b6 <> b11 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b7 <> b11 &
b8 <> b9 &
b8 <> b10 &
b8 <> b11 &
b9 <> b10 &
b9 <> b11 &
b10 <> b11 holds
CompF b3,
b2 = ((((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b11
theorem Th36: :: BVFUNC24:36
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10,
b11 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b3 <> b11 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b4 <> b11 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b5 <> b11 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b6 <> b11 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b7 <> b11 &
b8 <> b9 &
b8 <> b10 &
b8 <> b11 &
b9 <> b10 &
b9 <> b11 &
b10 <> b11 holds
CompF b4,
b2 = ((((((b3 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b11
theorem Th37: :: BVFUNC24:37
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10,
b11 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b3 <> b11 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b4 <> b11 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b5 <> b11 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b6 <> b11 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b7 <> b11 &
b8 <> b9 &
b8 <> b10 &
b8 <> b11 &
b9 <> b10 &
b9 <> b11 &
b10 <> b11 holds
CompF b5,
b2 = ((((((b3 '/\' b4) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b11
theorem Th38: :: BVFUNC24:38
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10,
b11 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b3 <> b11 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b4 <> b11 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b5 <> b11 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b6 <> b11 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b7 <> b11 &
b8 <> b9 &
b8 <> b10 &
b8 <> b11 &
b9 <> b10 &
b9 <> b11 &
b10 <> b11 holds
CompF b6,
b2 = ((((((b3 '/\' b4) '/\' b5) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b11
theorem Th39: :: BVFUNC24:39
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10,
b11 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b3 <> b11 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b4 <> b11 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b5 <> b11 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b6 <> b11 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b7 <> b11 &
b8 <> b9 &
b8 <> b10 &
b8 <> b11 &
b9 <> b10 &
b9 <> b11 &
b10 <> b11 holds
CompF b7,
b2 = ((((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b8) '/\' b9) '/\' b10) '/\' b11
theorem Th40: :: BVFUNC24:40
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10,
b11 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b3 <> b11 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b4 <> b11 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b5 <> b11 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b6 <> b11 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b7 <> b11 &
b8 <> b9 &
b8 <> b10 &
b8 <> b11 &
b9 <> b10 &
b9 <> b11 &
b10 <> b11 holds
CompF b8,
b2 = ((((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b9) '/\' b10) '/\' b11
theorem Th41: :: BVFUNC24:41
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10,
b11 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b3 <> b11 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b4 <> b11 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b5 <> b11 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b6 <> b11 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b7 <> b11 &
b8 <> b9 &
b8 <> b10 &
b8 <> b11 &
b9 <> b10 &
b9 <> b11 &
b10 <> b11 holds
CompF b9,
b2 = ((((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b10) '/\' b11
theorem Th42: :: BVFUNC24:42
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10,
b11 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b3 <> b11 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b4 <> b11 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b5 <> b11 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b6 <> b11 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b7 <> b11 &
b8 <> b9 &
b8 <> b10 &
b8 <> b11 &
b9 <> b10 &
b9 <> b11 &
b10 <> b11 holds
CompF b10,
b2 = ((((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b11
theorem Th43: :: BVFUNC24:43
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10,
b11 being
a_partition of
b1 st
b2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b3 <> b11 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b4 <> b11 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b5 <> b11 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b6 <> b11 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b7 <> b11 &
b8 <> b9 &
b8 <> b10 &
b8 <> b11 &
b9 <> b10 &
b9 <> b11 &
b10 <> b11 holds
CompF b11,
b2 = ((((((b3 '/\' b4) '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10
theorem Th44: :: BVFUNC24:44
for
b1,
b2,
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
set for
b10 being
Functionfor
b11,
b12,
b13,
b14,
b15,
b16,
b17,
b18,
b19 being
set st
b1 <> b2 &
b1 <> b3 &
b1 <> b4 &
b1 <> b5 &
b1 <> b6 &
b1 <> b7 &
b1 <> b8 &
b1 <> b9 &
b2 <> b3 &
b2 <> b4 &
b2 <> b5 &
b2 <> b6 &
b2 <> b7 &
b2 <> b8 &
b2 <> b9 &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b7 <> b8 &
b7 <> b9 &
b8 <> b9 &
b10 = ((((((((b2 .--> b12) +* (b3 .--> b13)) +* (b4 .--> b14)) +* (b5 .--> b15)) +* (b6 .--> b16)) +* (b7 .--> b17)) +* (b8 .--> b18)) +* (b9 .--> b19)) +* (b1 .--> b11) holds
(
b10 . b1 = b11 &
b10 . b2 = b12 &
b10 . b3 = b13 &
b10 . b4 = b14 &
b10 . b5 = b15 &
b10 . b6 = b16 &
b10 . b7 = b17 &
b10 . b8 = b18 &
b10 . b9 = b19 )
theorem Th45: :: BVFUNC24:45
for
b1,
b2,
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
set for
b10 being
Functionfor
b11,
b12,
b13,
b14,
b15,
b16,
b17,
b18,
b19 being
set st
b10 = ((((((((b2 .--> b12) +* (b3 .--> b13)) +* (b4 .--> b14)) +* (b5 .--> b15)) +* (b6 .--> b16)) +* (b7 .--> b17)) +* (b8 .--> b18)) +* (b9 .--> b19)) +* (b1 .--> b11) holds
dom b10 = {b1,b2,b3,b4,b5,b6,b7,b8,b9}
theorem Th46: :: BVFUNC24:46
for
b1,
b2,
b3,
b4,
b5,
b6,
b7,
b8,
b9 being
set for
b10 being
Functionfor
b11,
b12,
b13,
b14,
b15,
b16,
b17,
b18,
b19 being
set st
b10 = ((((((((b2 .--> b12) +* (b3 .--> b13)) +* (b4 .--> b14)) +* (b5 .--> b15)) +* (b6 .--> b16)) +* (b7 .--> b17)) +* (b8 .--> b18)) +* (b9 .--> b19)) +* (b1 .--> b11) holds
rng b10 = {(b10 . b1),(b10 . b2),(b10 . b3),(b10 . b4),(b10 . b5),(b10 . b6),(b10 . b7),(b10 . b8),(b10 . b9)}
theorem Th47: :: BVFUNC24:47
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10,
b11 being
a_partition of
b1for
b12,
b13 being
Element of
b1 st
b2 is
independent &
b2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b3 <> b11 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b4 <> b11 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b5 <> b11 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b6 <> b11 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b7 <> b11 &
b8 <> b9 &
b8 <> b10 &
b8 <> b11 &
b9 <> b10 &
b9 <> b11 &
b10 <> b11 holds
(EqClass b13,(((((((b4 '/\' b5) '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b11)) /\ (EqClass b12,b3) <> {}
theorem Th48: :: BVFUNC24:48
for
b1 being non
empty set for
b2 being
Subset of
(PARTITIONS b1)for
b3,
b4,
b5,
b6,
b7,
b8,
b9,
b10,
b11 being
a_partition of
b1for
b12,
b13 being
Element of
b1 st
b2 is
independent &
b2 = {b3,b4,b5,b6,b7,b8,b9,b10,b11} &
b3 <> b4 &
b3 <> b5 &
b3 <> b6 &
b3 <> b7 &
b3 <> b8 &
b3 <> b9 &
b3 <> b10 &
b3 <> b11 &
b4 <> b5 &
b4 <> b6 &
b4 <> b7 &
b4 <> b8 &
b4 <> b9 &
b4 <> b10 &
b4 <> b11 &
b5 <> b6 &
b5 <> b7 &
b5 <> b8 &
b5 <> b9 &
b5 <> b10 &
b5 <> b11 &
b6 <> b7 &
b6 <> b8 &
b6 <> b9 &
b6 <> b10 &
b6 <> b11 &
b7 <> b8 &
b7 <> b9 &
b7 <> b10 &
b7 <> b11 &
b8 <> b9 &
b8 <> b10 &
b8 <> b11 &
b9 <> b10 &
b9 <> b11 &
b10 <> b11 &
EqClass b12,
((((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b11) = EqClass b13,
((((((b5 '/\' b6) '/\' b7) '/\' b8) '/\' b9) '/\' b10) '/\' b11) holds
EqClass b13,
(CompF b3,b2) meets EqClass b12,
(CompF b4,b2)