let A be non empty finite set ; :: thesis: for L being Function of (bool A),(bool A) st L . A = A & ( for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ) holds

ex R being non empty finite RelStr st

( the carrier of R = A & L = LAp R )

let L be Function of (bool A),(bool A); :: thesis: ( L . A = A & ( for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ) implies ex R being non empty finite RelStr st

( the carrier of R = A & L = LAp R ) )

assume that

A1: L . A = A and

A2: for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ; :: thesis: ex R being non empty finite RelStr st

( the carrier of R = A & L = LAp R )

set U = Flip L;

A3: (Flip L) . {} = {} by Th19, A1;

A4: for X, Y being Subset of A holds (Flip L) . (X \/ Y) = ((Flip L) . X) \/ ((Flip L) . Y) by Th22, A2;

consider R being non empty finite RelStr such that

A5: ( the carrier of R = A & Flip L = UAp R ) by Th29, A3, A4;

take R ; :: thesis: ( the carrier of R = A & L = LAp R )

L = Flip (UAp R) by Th23, A5;

hence ( the carrier of R = A & L = LAp R ) by A5, Th27; :: thesis: verum

ex R being non empty finite RelStr st

( the carrier of R = A & L = LAp R )

let L be Function of (bool A),(bool A); :: thesis: ( L . A = A & ( for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ) implies ex R being non empty finite RelStr st

( the carrier of R = A & L = LAp R ) )

assume that

A1: L . A = A and

A2: for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ; :: thesis: ex R being non empty finite RelStr st

( the carrier of R = A & L = LAp R )

set U = Flip L;

A3: (Flip L) . {} = {} by Th19, A1;

A4: for X, Y being Subset of A holds (Flip L) . (X \/ Y) = ((Flip L) . X) \/ ((Flip L) . Y) by Th22, A2;

consider R being non empty finite RelStr such that

A5: ( the carrier of R = A & Flip L = UAp R ) by Th29, A3, A4;

take R ; :: thesis: ( the carrier of R = A & L = LAp R )

L = Flip (UAp R) by Th23, A5;

hence ( the carrier of R = A & L = LAp R ) by A5, Th27; :: thesis: verum