Article PARTIT_2, MML version 4.99.1005
:: PARTIT_2:modenot 1 => PARTIT_2:mode 1
definition
let a1 be non empty set;
let a2 be non empty Element of bool PARTITIONS a1;
redefine mode Element of a2 -> a_partition of a1;
end;
:: PARTIT_2:th 1
theorem
for b1 being non empty set holds
'/\' {} PARTITIONS b1 = %O b1;
:: PARTIT_2:th 2
theorem
for b1 being non empty set
for b2, b3 being symmetric transitive total Relation of b1,b1 holds
b2 \/ b3 c= b2 * b3;
:: PARTIT_2:th 3
theorem
for b1 being non empty set
for b2 being Relation of b1,b1 holds
b2 c= nabla b1;
:: PARTIT_2:th 4
theorem
for b1 being non empty set
for b2 being symmetric transitive total Relation of b1,b1 holds
(nabla b1) * b2 = nabla b1 & b2 * nabla b1 = nabla b1;
:: PARTIT_2:th 5
theorem
for b1 being non empty set
for b2 being a_partition of b1
for b3, b4 being Element of b1 holds
[b3,b4] in ERl b2
iff
b3 in EqClass(b4,b2);
:: PARTIT_2:th 6
theorem
for b1 being non empty set
for b2, b3, b4 being a_partition of b1
st ERl b4 = (ERl b2) * ERl b3
for b5, b6 being Element of b1 holds
b5 in EqClass(b6,b4)
iff
ex b7 being Element of b1 st
b5 in EqClass(b7,b2) & b7 in EqClass(b6,b3);
:: PARTIT_2:th 7
theorem
for b1, b2 being Relation-like set
for b3 being set
st b1 is_reflexive_in b3 & b2 is_reflexive_in b3
holds b1 * b2 is_reflexive_in b3;
:: PARTIT_2:th 8
theorem
for b1 being Relation-like set
for b2 being set
st b1 is_reflexive_in b2
holds b2 c= field b1;
:: PARTIT_2:th 9
theorem
for b1 being set
for b2 being Relation of b1,b1
st b2 is_reflexive_in b1
holds b1 = field b2;
:: PARTIT_2:th 10
theorem
for b1 being non empty set
for b2, b3 being symmetric transitive total Relation of b1,b1
st b2 * b3 = b3 * b2
holds b2 * b3 is symmetric transitive total Relation of b1,b1;
:: PARTIT_2:th 11
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
st b2 '<' b3
holds 'not' b3 '<' 'not' b2;
:: PARTIT_2:th 13
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
for b4 being Element of bool PARTITIONS b1
for b5 being a_partition of b1
st b2 '<' b3
holds All(b2,b5,b4) '<' All(b3,b5,b4);
:: PARTIT_2:th 15
theorem
for b1 being non empty set
for b2, b3 being Element of Funcs(b1,BOOLEAN)
for b4 being Element of bool PARTITIONS b1
for b5 being a_partition of b1
st b2 '<' b3
holds Ex(b2,b5,b4) '<' Ex(b3,b5,b4);
:: PARTIT_2:th 16
theorem
for b1 being non empty set
for b2 being Element of bool PARTITIONS b1
st b2 is independent(b1)
for b3, b4 being Element of bool PARTITIONS b1
st b3 c= b2 & b4 c= b2
holds (ERl '/\' b3) * ERl '/\' b4 = (ERl '/\' b4) * ERl '/\' b3;
:: PARTIT_2:th 17
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds All(All(b2,b4,b3),b5,b3) = All(All(b2,b5,b3),b4,b3);
:: PARTIT_2:th 18
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex(Ex(b2,b4,b3),b5,b3) = Ex(Ex(b2,b5,b3),b4,b3);
:: PARTIT_2:th 19
theorem
for b1 being non empty set
for b2 being Element of Funcs(b1,BOOLEAN)
for b3 being Element of bool PARTITIONS b1
for b4, b5 being a_partition of b1
st b3 is independent(b1)
holds Ex(All(b2,b4,b3),b5,b3) '<' All(Ex(b2,b5,b3),b4,b3);