let it1, it2 be BinOp of (BoundedLinearOperators (X,X)); :: thesis: ( ( for f, g being Element of BoundedLinearOperators (X,X) holds it1 . (f,g) = f * g ) & ( for f, g being Element of BoundedLinearOperators (X,X) holds it2 . (f,g) = f * g ) implies it1 = it2 )
assume that
A2: for f, g being Element of BoundedLinearOperators (X,X) holds it1 . (f,g) = f * g and
A3: for f, g being Element of BoundedLinearOperators (X,X) holds it2 . (f,g) = f * g ; :: thesis: it1 = it2

now :: thesis: for f, g being Element of BoundedLinearOperators (X,X) holds it1 . (f,g) = it2 . (f,g)

let f, g be Element of BoundedLinearOperators (X,X); :: thesis: it1 . (f,g) = it2 . (f,g)
thus it1 . (f,g) = f * g by A2
.= it2 . (f,g) by A3 ; :: thesis: verum

end;

hence it1 = it2 ; :: thesis: verum