let f1, f2 be preRIF of F₁(); :: thesis: ( ( for x, y being Subset of F₁() holds f1 . (x,y) = F₂(x,y) ) & ( for x, y being Subset of F₁() holds f2 . (x,y) = F₂(x,y) ) implies f1 = f2 )
assume that
A1: for x, y being Subset of F₁() holds f1 . (x,y) = F₂(x,y) and
A2: for x, y being Subset of F₁() holds f2 . (x,y) = F₂(x,y) ; :: thesis: f1 = f2
for x, y being Element of bool the carrier of F₁() holds f2 . (x,y) = f1 . (x,y)

proof

let x, y be Element of bool the carrier of F₁(); :: thesis: f2 . (x,y) = f1 . (x,y)
f1 . (x,y) = F₂(x,y) by A1
.= f2 . (x,y) by A2 ;
hence f2 . (x,y) = f1 . (x,y) ; :: thesis: verum

end;

hence f1 = f2 by BINOP_1:2; :: thesis: verum