let f1, f2 be BinOp of [.0,1.]; :: thesis: ( ( for a, b being Element of [.0,1.] holds f1 . (a,b) = (a * b) / ((a + b) - (a * b)) ) & ( for a, b being Element of [.0,1.] holds f2 . (a,b) = (a * b) / ((a + b) - (a * b)) ) implies f1 = f2 )
assume that
A1: for a, b being Element of [.0,1.] holds f1 . (a,b) = (a * b) / ((a + b) - (a * b)) and
A2: for a, b being Element of [.0,1.] holds f2 . (a,b) = (a * b) / ((a + b) - (a * b)) ; :: thesis: f1 = f2
for a, b being set st a in [.0,1.] & b in [.0,1.] holds
f1 . (a,b) = f2 . (a,b)

proof

let a, b be set ; :: thesis: ( a in [.0,1.] & b in [.0,1.] implies f1 . (a,b) = f2 . (a,b) )
assume ( a in [.0,1.] & b in [.0,1.] ) ; :: thesis: f1 . (a,b) = f2 . (a,b)
then reconsider aa = a, bb = b as Element of [.0,1.] ;
f1 . (aa,bb) = (aa * bb) / ((aa + bb) - (aa * bb)) by A1
.= f2 . (aa,bb) by A2 ;
hence f1 . (a,b) = f2 . (a,b) ; :: thesis: verum

end;

hence f1 = f2 by BINOP_1:def 21; :: thesis: verum