let f1, f2 be BinOp of NAT; :: thesis: ( ( for m, n being Nat holds f1 . (m,n) = m gcd n ) & ( for m, n being Nat holds f2 . (m,n) = m gcd n ) implies f1 = f2 )
assume that
A2: for m, n being Nat holds f1 . (m,n) = m gcd n and
A3: for m, n being Nat holds f2 . (m,n) = m gcd n ; :: thesis: f1 = f2

now

let m, n be Element of NAT ; :: thesis: f1 . (m,n) = f2 . (m,n)
thus f1 . (m,n) = m gcd n by A2
.= f2 . (m,n) by A3 ; :: thesis: verum

end;

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