Article BVFUNC24, MML version 4.99.1005
:: BVFUNC24:th 1
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 2
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 3
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 4
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 5
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 6
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 7
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 8
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Relation-like Function-like set
for 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;
:: BVFUNC24:th 9
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Relation-like Function-like set
for 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 proj1 b8 = {b1,b2,b3,b4,b5,b6,b7};
:: BVFUNC24:th 10
theorem
for b1, b2, b3, b4, b5, b6, b7 being set
for b8 being Relation-like Function-like set
for 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 proj2 b8 = {b8 . b1,b8 . b2,b8 . b3,b8 . b4,b8 . b5,b8 . b6,b8 . b7};
:: BVFUNC24:th 11
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8, b9 being a_partition of b1
for b10, b11 being Element of b1
st b2 is independent(b1) & 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);
:: BVFUNC24:th 12
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8, b9 being a_partition of b1
for b10, b11 being Element of b1
st b2 is independent(b1) &
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));
:: BVFUNC24:th 13
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 14
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 15
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 16
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 17
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 18
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 19
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 20
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 21
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Relation-like Function-like set
for 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;
:: BVFUNC24:th 22
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Relation-like Function-like set
for 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 proj1 b9 = {b1,b2,b3,b4,b5,b6,b7,b8};
:: BVFUNC24:th 23
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being set
for b9 being Relation-like Function-like set
for 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 proj2 b9 = {b9 . b1,b9 . b2,b9 . b3,b9 . b4,b9 . b5,b9 . b6,b9 . b7,b9 . b8};
:: BVFUNC24:th 24
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8, b9, b10 being a_partition of b1
for b11, b12 being Element of b1
st b2 is independent(b1) & 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) <> {};
:: BVFUNC24:th 25
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8, b9, b10 being a_partition of b1
for b11, b12 being Element of b1
st b2 is independent(b1) &
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));
:: BVFUNC24:th 35
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 36
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 37
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 38
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 39
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 40
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 41
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 42
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 43
theorem
for b1 being non empty set
for b2 being Element of bool 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;
:: BVFUNC24:th 44
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Relation-like Function-like set
for 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;
:: BVFUNC24:th 45
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Relation-like Function-like set
for 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 proj1 b10 = {b1,b2,b3,b4,b5,b6,b7,b8,b9};
:: BVFUNC24:th 46
theorem
for b1, b2, b3, b4, b5, b6, b7, b8, b9 being set
for b10 being Relation-like Function-like set
for 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 proj2 b10 = {b10 . b1,b10 . b2,b10 . b3,b10 . b4,b10 . b5,b10 . b6,b10 . b7,b10 . b8,b10 . b9};
:: BVFUNC24:th 47
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8, b9, b10, b11 being a_partition of b1
for b12, b13 being Element of b1
st b2 is independent(b1) & 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) <> {};
:: BVFUNC24:th 48
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
for b3, b4, b5, b6, b7, b8, b9, b10, b11 being a_partition of b1
for b12, b13 being Element of b1
st b2 is independent(b1) &
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));