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

now :: thesis: for v being Vector of V
for w being Vector of W holds F . (v,w) = G . (v,w)

let v be Vector of V; :: thesis: for w being Vector of W holds F . (v,w) = G . (v,w)
let w be Vector of W; :: thesis: F . (v,w) = G . (v,w)
thus F . (v,w) = (f . (v,w)) + (g . (v,w)) by A2
.= G . (v,w) by A3 ; :: thesis: verum

end;

hence F = G ; :: thesis: verum