deffunc H₁( Subset of R, Subset of R) -> set = (((CMap (kappa R)) . ($1,$2)) + ((CMap (kappa R)) . ($2,$1))) / 2;
let f1, f2 be preRIF of R; :: thesis: ( ( for x, y being Subset of R holds f1 . (x,y) = (((CMap (kappa R)) . (x,y)) + ((CMap (kappa R)) . (y,x))) / 2 ) & ( for x, y being Subset of R holds f2 . (x,y) = (((CMap (kappa R)) . (x,y)) + ((CMap (kappa R)) . (y,x))) / 2 ) implies f1 = f2 )
assume that
A1: for x, y being Subset of R holds f1 . (x,y) = H₁(x,y) and
A2: for x, y being Subset of R holds f2 . (x,y) = H₁(x,y) ; :: thesis: f1 = f2
for x, y being Subset of R holds f1 . (x,y) = f2 . (x,y)

proof

let x, y be Subset of R; :: thesis: f1 . (x,y) = f2 . (x,y)
f1 . (x,y) = H₁(x,y) by A1
.= f2 . (x,y) by A2 ;
hence f1 . (x,y) = f2 . (x,y) ; :: thesis: verum

end;

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