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