set L = LKer f;
set vq = VectQuot (V,(LKer f));
let f1, f2 be additiveSAF homogeneousSAF Form of (VectQuot (V,(LKer f))),W; :: thesis: ( ( for A being Vector of (VectQuot (V,(LKer f)))
for w being Vector of W
for v being Vector of V st A = v + (LKer f) holds
f1 . (A,w) = f . (v,w) ) & ( for A being Vector of (VectQuot (V,(LKer f)))
for w being Vector of W
for v being Vector of V st A = v + (LKer f) holds
f2 . (A,w) = f . (v,w) ) implies f1 = f2 )
assume that
A13: for A being Vector of (VectQuot (V,(LKer f)))
for w being Vector of W
for a being Vector of V st A = a + (LKer f) holds
f1 . (A,w) = f . (a,w) and
A14: for A being Vector of (VectQuot (V,(LKer f)))
for w being Vector of W
for a being Vector of V st A = a + (LKer f) holds
f2 . (A,w) = f . (a,w) ; :: thesis: f1 = f2

now :: thesis: for A being Vector of (VectQuot (V,(LKer f)))
for w being Vector of W holds f1 . (A,w) = f2 . (A,w)

let A be Vector of (VectQuot (V,(LKer f))); :: thesis: for w being Vector of W holds f1 . (A,w) = f2 . (A,w)
let w be Vector of W; :: thesis: f1 . (A,w) = f2 . (A,w)
consider a being Vector of V such that
A15: A = a + (LKer f) by VECTSP10:22;
thus f1 . (A,w) = f . (a,w) by A13, A15
.= f2 . (A,w) by A14, A15 ; :: thesis: verum

end;

hence f1 = f2 ; :: thesis: verum