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

assume that

A2: for n, m being Nat holds f1 . (m,n) = m gcd n and

A3: for n, m being Nat holds f2 . (m,n) = m gcd n ; :: thesis: f1 = f2

assume that

A2: for n, m being Nat holds f1 . (m,n) = m gcd n and

A3: for n, m being Nat holds f2 . (m,n) = m gcd n ; :: thesis: f1 = f2

now :: thesis: for m, n being Element of NAT holds f1 . (m,n) = f2 . (m,n)

hence
f1 = f2
; :: thesis: verumlet 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;thus f1 . (m,n) = m gcd n by A2

.= f2 . (m,n) by A3 ; :: thesis: verum