let h be Form of V,W; :: thesis: ( h = f - g iff for v being Vector of V
for w being Vector of W holds h . (v,w) = (f . (v,w)) - (g . (v,w)) )
thus ( h = f - g implies for v being Vector of V
for w being Vector of W holds h . (v,w) = (f . (v,w)) - (g . (v,w)) ) :: thesis: ( ( for v being Vector of V
for w being Vector of W holds h . (v,w) = (f . (v,w)) - (g . (v,w)) ) implies h = f - g )

proof

assume A1: h = f - g ; :: thesis: for v being Vector of V
for w being Vector of W holds h . (v,w) = (f . (v,w)) - (g . (v,w))
let v be Vector of V; :: thesis: for w being Vector of W holds h . (v,w) = (f . (v,w)) - (g . (v,w))
let w be Vector of W; :: thesis: h . (v,w) = (f . (v,w)) - (g . (v,w))
thus h . (v,w) = (f . (v,w)) + ((- g) . (v,w)) by A1, BLDef2
.= (f . (v,w)) - (g . (v,w)) by BLDef4 ; :: thesis: verum

end;

assume A2: for v being Vector of V
for w being Vector of W holds h . (v,w) = (f . (v,w)) - (g . (v,w)) ; :: thesis: h = f - g

now :: thesis: for v being Vector of V
for w being Vector of W holds h . (v,w) = (f - g) . (v,w)

let v be Vector of V; :: thesis: for w being Vector of W holds h . (v,w) = (f - g) . (v,w)
let w be Vector of W; :: thesis: h . (v,w) = (f - g) . (v,w)
thus h . (v,w) = (f . (v,w)) - (g . (v,w)) by A2
.= (f . (v,w)) + ((- g) . (v,w)) by BLDef4
.= (f - g) . (v,w) by BLDef2 ; :: thesis: verum

end;

hence h = f - g ; :: thesis: verum