let F1, F2 be Form of V,W; :: thesis: ( ( for v being Vector of V
for w being Vector of W holds F1 . (v,w) = (f . v) * (g . w) ) & ( for v being Vector of V
for w being Vector of W holds F2 . (v,w) = (f . v) * (g . w) ) implies F1 = F2 )
assume that
A2: for v being Vector of V
for y being Vector of W holds F1 . (v,y) = (f . v) * (g . y) and
A3: for v being Vector of V
for y being Vector of W holds F2 . (v,y) = (f . v) * (g . y) ; :: thesis: F1 = F2

now :: thesis: for v being Vector of V
for y being Vector of W holds F1 . (v,y) = F2 . (v,y)

let v be Vector of V; :: thesis: for y being Vector of W holds F1 . (v,y) = F2 . (v,y)
let y be Vector of W; :: thesis: F1 . (v,y) = F2 . (v,y)
thus F1 . (v,y) = (f . v) * (g . y) by A2
.= F2 . (v,y) by A3 ; :: thesis: verum

end;

hence F1 = F2 ; :: thesis: verum