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 . X) ` c= L . ((L . X) `) ) & ( for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ) holds

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

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

assume that

A1: L . A = A and

A3: for X being Subset of A holds (L . X) ` c= L . ((L . X) `) and

A4: for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ; :: thesis: ex R being non empty finite negative_alliance RelStr st

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

set U = Flip L;

C1: (Flip L) . {} = {} by A1, ROUGHS_2:19;

C2: for X being Subset of A holds (Flip L) . (((Flip L) . X) `) c= ((Flip L) . X) ` by A3, Conv2;

for X, Y being Subset of A holds (Flip L) . (X \/ Y) = ((Flip L) . X) \/ ((Flip L) . Y) by A4, ROUGHS_2:22;

then consider R being non empty finite negative_alliance RelStr such that

A2: ( the carrier of R = A & Flip L = UAp R ) by Prop14, C1, C2;

Flip (Flip L) = LAp R by A2, ROUGHS_2:27;

then L = LAp R by ROUGHS_2:23;

hence ex R being non empty finite negative_alliance RelStr st

( the carrier of R = A & L = LAp R ) by A2; :: thesis: verum

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

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

assume that

A1: L . A = A and

A3: for X being Subset of A holds (L . X) ` c= L . ((L . X) `) and

A4: for X, Y being Subset of A holds L . (X /\ Y) = (L . X) /\ (L . Y) ; :: thesis: ex R being non empty finite negative_alliance RelStr st

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

set U = Flip L;

C1: (Flip L) . {} = {} by A1, ROUGHS_2:19;

C2: for X being Subset of A holds (Flip L) . (((Flip L) . X) `) c= ((Flip L) . X) ` by A3, Conv2;

for X, Y being Subset of A holds (Flip L) . (X \/ Y) = ((Flip L) . X) \/ ((Flip L) . Y) by A4, ROUGHS_2:22;

then consider R being non empty finite negative_alliance RelStr such that

A2: ( the carrier of R = A & Flip L = UAp R ) by Prop14, C1, C2;

Flip (Flip L) = LAp R by A2, ROUGHS_2:27;

then L = LAp R by ROUGHS_2:23;

hence ex R being non empty finite negative_alliance RelStr st

( the carrier of R = A & L = LAp R ) by A2; :: thesis: verum