set L = RKer f;
set vq = VectQuot (W,(RKer f));
set C = CosetSet (W,(RKer f));
set aC = addCoset (W,(RKer f));
set mC = lmultCoset (W,(RKer f));
defpred S₁[ set , set , set ] means for w being Vector of W st $2 = w + (RKer f) holds
$3 = f . ($1,w);

A1: now :: thesis: for v0 being Vector of V
for A being Vector of (VectQuot (W,(RKer f))) ex y being Element of the carrier of K st S₁[v0,A,y]

let v0 be Vector of V; :: thesis: for A being Vector of (VectQuot (W,(RKer f))) ex y being Element of the carrier of K st S₁[v0,A,y]
let A be Vector of (VectQuot (W,(RKer f))); :: thesis: ex y being Element of the carrier of K st S₁[v0,A,y]
consider a being Vector of W such that
A2: A = a + (RKer f) by VECTSP10:22;
take y = f . (v0,a); :: thesis: S₁[v0,A,y]

now :: thesis: for a1 being Vector of W st A = a1 + (RKer f) holds
y = f . (v0,a1)

let a1 be Vector of W; :: thesis: ( A = a1 + (RKer f) implies y = f . (v0,a1) )
assume A = a1 + (RKer f) ; :: thesis: y = f . (v0,a1)
then a in a1 + (RKer f) by A2, VECTSP_4:44;
then consider w being Vector of W such that
A3: w in RKer f and
A4: a = a1 + w by VECTSP_4:42;
w in the carrier of (RKer f) by A3;
then w in rightker f by Def19;
then A5: ex aa being Vector of W st
( aa = w & ( for vv being Vector of V holds f . (vv,aa) = 0. K ) ) ;
thus y = (f . (v0,a1)) + (f . (v0,w)) by A4, Th27
.= (f . (v0,a1)) + (0. K) by A5
.= f . (v0,a1) by RLVECT_1:def 4 ; :: thesis: verum

end;

hence S₁[v0,A,y] ; :: thesis: verum

end;

consider ff being Function of [: the carrier of V, the carrier of (VectQuot (W,(RKer f))):], the carrier of K such that
A6: for v being Vector of V
for A being Vector of (VectQuot (W,(RKer f))) holds S₁[v,A,ff . (v,A)] from BINOP_1:sch 3(A1);
reconsider ff = ff as Form of V,(VectQuot (W,(RKer f))) ;
A7: CosetSet (W,(RKer f)) = the carrier of (VectQuot (W,(RKer f))) by VECTSP10:def 6;

now :: thesis: for v being Vector of V holds FunctionalFAF (ff,v) is additive

let v be Vector of V; :: thesis: FunctionalFAF (ff,v) is additive
set ffw = FunctionalFAF (ff,v);

now :: thesis: for A, B being Vector of (VectQuot (W,(RKer f))) holds (FunctionalFAF (ff,v)) . (A + B) = ((FunctionalFAF (ff,v)) . A) + ((FunctionalFAF (ff,v)) . B)

let A, B be Vector of (VectQuot (W,(RKer f))); :: thesis: (FunctionalFAF (ff,v)) . (A + B) = ((FunctionalFAF (ff,v)) . A) + ((FunctionalFAF (ff,v)) . B)
consider a being Vector of W such that
A8: A = a + (RKer f) by VECTSP10:22;
consider b being Vector of W such that
A9: B = b + (RKer f) by VECTSP10:22;
A10: ( the addF of (VectQuot (W,(RKer f))) = addCoset (W,(RKer f)) & (addCoset (W,(RKer f))) . (A,B) = (a + b) + (RKer f) ) by A7, A8, A9, VECTSP10:def 3, VECTSP10:def 6;
thus (FunctionalFAF (ff,v)) . (A + B) = ff . (v,(A + B)) by Th8
.= f . (v,(a + b)) by A6, A10, RLVECT_1:2
.= (f . (v,a)) + (f . (v,b)) by Th27
.= (ff . (v,A)) + (f . (v,b)) by A6, A8
.= (ff . (v,A)) + (ff . (v,B)) by A6, A9
.= ((FunctionalFAF (ff,v)) . A) + (ff . (v,B)) by Th8
.= ((FunctionalFAF (ff,v)) . A) + ((FunctionalFAF (ff,v)) . B) by Th8 ; :: thesis: verum

end;

hence FunctionalFAF (ff,v) is additive ; :: thesis: verum

end;

then reconsider ff = ff as additiveFAF Form of V,(VectQuot (W,(RKer f))) by Def11;

now :: thesis: for v being Vector of V holds FunctionalFAF (ff,v) is homogeneous

let v be Vector of V; :: thesis: FunctionalFAF (ff,v) is homogeneous
set ffw = FunctionalFAF (ff,v);

now :: thesis: for A being Vector of (VectQuot (W,(RKer f)))
for r being Element of K holds (FunctionalFAF (ff,v)) . (r * A) = r * ((FunctionalFAF (ff,v)) . A)

let A be Vector of (VectQuot (W,(RKer f))); :: thesis: for r being Element of K holds (FunctionalFAF (ff,v)) . (r * A) = r * ((FunctionalFAF (ff,v)) . A)
let r be Element of K; :: thesis: (FunctionalFAF (ff,v)) . (r * A) = r * ((FunctionalFAF (ff,v)) . A)
consider a being Vector of W such that
A11: A = a + (RKer f) by VECTSP10:22;
A12: ( the lmult of (VectQuot (W,(RKer f))) = lmultCoset (W,(RKer f)) & (lmultCoset (W,(RKer f))) . (r,A) = (r * a) + (RKer f) ) by A7, A11, VECTSP10:def 5, VECTSP10:def 6;
thus (FunctionalFAF (ff,v)) . (r * A) = ff . (v,(r * A)) by Th8
.= f . (v,(r * a)) by A6, A12
.= r * (f . (v,a)) by Th32
.= r * (ff . (v,A)) by A6, A11
.= r * ((FunctionalFAF (ff,v)) . A) by Th8 ; :: thesis: verum

end;

hence FunctionalFAF (ff,v) is homogeneous ; :: thesis: verum

end;

then reconsider ff = ff as additiveFAF homogeneousFAF Form of V,(VectQuot (W,(RKer f))) by Def13;
take ff ; :: 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
ff . (v,A) = f . (v,w)
thus 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
ff . (v,A) = f . (v,w) by A6; :: thesis: verum