set lf = RQForm f;
thus RQForm f is additiveSAF :: thesis: RQForm f is homogeneousSAF

proof

let A be Vector of (VectQuot W,(RKer f)); :: according to BILINEAR:def 13 :: thesis: FunctionalSAF (RQForm f),A is additive
set flf = FunctionalSAF (RQForm f),A;
consider w being Vector of W such that
A3: A = w + (RKer f) by VECTSP10:23;
let v, t be Vector of V; :: according to GRCAT_1:def 13 :: thesis: (FunctionalSAF (RQForm f),A) . (v + t) = ((FunctionalSAF (RQForm f),A) . v) + ((FunctionalSAF (RQForm f),A) . t)
thus (FunctionalSAF (RQForm f),A) . (v + t) = (RQForm f) . (v + t),A by Th10
.= f . (v + t),w by A3, Def22
.= (f . v,w) + (f . t,w) by Th27
.= ((RQForm f) . v,A) + (f . t,w) by A3, Def22
.= ((RQForm f) . v,A) + ((RQForm f) . t,A) by A3, Def22
.= ((FunctionalSAF (RQForm f),A) . v) + ((RQForm f) . t,A) by Th10
.= ((FunctionalSAF (RQForm f),A) . v) + ((FunctionalSAF (RQForm f),A) . t) by Th10 ; :: thesis: verum

end;

let A be Vector of (VectQuot W,(RKer f)); :: according to BILINEAR:def 15 :: thesis: FunctionalSAF (RQForm f),A is homogeneous
set flf = FunctionalSAF (RQForm f),A;
consider w being Vector of W such that
A4: A = w + (RKer f) by VECTSP10:23;
let v be Vector of V; :: according to HAHNBAN1:def 12 :: thesis: for b₁ being Element of the carrier of K holds (FunctionalSAF (RQForm f),A) . (b₁ * v) = b₁ * ((FunctionalSAF (RQForm f),A) . v)
let r be Scalar of ; :: thesis: (FunctionalSAF (RQForm f),A) . (r * v) = r * ((FunctionalSAF (RQForm f),A) . v)
thus (FunctionalSAF (RQForm f),A) . (r * v) = (RQForm f) . (r * v),A by Th10
.= f . (r * v),w by A4, Def22
.= r * (f . v,w) by Th32
.= r * ((RQForm f) . v,A) by A4, Def22
.= r * ((FunctionalSAF (RQForm f),A) . v) by Th10 ; :: thesis: verum