deffunc H₁( Element of X, Element of X) -> set = (2 * (f . ($1,$2))) / (((f . ($1,p)) + (f . ($2,p))) + (f . ($1,$2)));
let f1, f2 be Function of [:X,X:],REAL; :: thesis: ( ( for x, y being Element of X holds f1 . (x,y) = (2 * (f . (x,y))) / (((f . (x,p)) + (f . (y,p))) + (f . (x,y))) ) & ( for x, y being Element of X holds f2 . (x,y) = (2 * (f . (x,y))) / (((f . (x,p)) + (f . (y,p))) + (f . (x,y))) ) implies f1 = f2 )
assume that
A1: for x, y being Element of X holds f1 . (x,y) = H₁(x,y) and
A2: for x, y being Element of X holds f2 . (x,y) = H₁(x,y) ; :: thesis: f1 = f2
for x, y being Element of X holds f1 . (x,y) = f2 . (x,y)

proof

let x, y be Element of X; :: 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