let it1, it2 be BinOp of (Funcs A,COMPLEX ); :: thesis: ( ( for f, g being Element of Funcs A,COMPLEX holds it1 . f,g = addcomplex .: f,g ) & ( for f, g being Element of Funcs A,COMPLEX holds it2 . f,g = addcomplex .: f,g ) implies it1 = it2 )
assume that
A2: for f, g being Element of Funcs A,COMPLEX holds it1 . f,g = addcomplex .: f,g and
A3: for f, g being Element of Funcs A,COMPLEX holds it2 . f,g = addcomplex .: f,g ; :: thesis: it1 = it2

now

let f, g be Element of Funcs A,COMPLEX ; :: thesis: it1 . f,g = it2 . f,g
thus it1 . f,g = addcomplex .: f,g by A2
.= it2 . f,g by A3 ; :: thesis: verum

end;

hence it1 = it2 by BINOP_1:2; :: thesis: verum