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

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

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

end;

hence f1 = f2 ; :: thesis: verum