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

ex R being non empty finite positive_alliance 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 being Subset of A holds L . ((L . X) `) c= (L . X) ` ) & ( for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ) implies ex R being non empty finite positive_alliance RelStr st

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

assume A1: ( L . A = A & ( for X being Subset of A holds L . ((L . X) `) c= (L . X) ` ) & ( for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ) ) ; :: thesis: ex R being non empty finite positive_alliance RelStr st

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

set U = Flip L;

A2: for X being Subset of A holds ((Flip L) . X) ` c= (Flip L) . (((Flip L) . X) `) by A1, Conv4;

A4: (Flip L) . {} = {} by A1, ROUGHS_2:19;

for X, Y being Subset of A holds (Flip L) . (X \/ Y) = ((Flip L) . X) \/ ((Flip L) . Y) by ROUGHS_2:22, A1;

then consider R being non empty finite positive_alliance RelStr such that

A3: ( the carrier of R = A & Flip L = UAp R ) by Prop11, A2, A4;

L = Flip (UAp R) by A3, ROUGHS_2:23;

then L = LAp R by ROUGHS_2:27;

hence ex R being non empty finite positive_alliance RelStr st

( the carrier of R = A & L = LAp R ) by A3; :: thesis: verum

ex R being non empty finite positive_alliance 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 being Subset of A holds L . ((L . X) `) c= (L . X) ` ) & ( for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ) implies ex R being non empty finite positive_alliance RelStr st

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

assume A1: ( L . A = A & ( for X being Subset of A holds L . ((L . X) `) c= (L . X) ` ) & ( for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ) ) ; :: thesis: ex R being non empty finite positive_alliance RelStr st

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

set U = Flip L;

A2: for X being Subset of A holds ((Flip L) . X) ` c= (Flip L) . (((Flip L) . X) `) by A1, Conv4;

A4: (Flip L) . {} = {} by A1, ROUGHS_2:19;

for X, Y being Subset of A holds (Flip L) . (X \/ Y) = ((Flip L) . X) \/ ((Flip L) . Y) by ROUGHS_2:22, A1;

then consider R being non empty finite positive_alliance RelStr such that

A3: ( the carrier of R = A & Flip L = UAp R ) by Prop11, A2, A4;

L = Flip (UAp R) by A3, ROUGHS_2:23;

then L = LAp R by ROUGHS_2:27;

hence ex R being non empty finite positive_alliance RelStr st

( the carrier of R = A & L = LAp R ) by A3; :: thesis: verum