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

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)

hence
F1 = F2
; :: thesis: verumfor 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;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