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

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 by BINOP_1:2; :: thesis: verum