
theorem Th34:
  for K be add-associative right_zeroed right_complementable
Abelian associative well-unital distributive non empty doubleLoopStr for V be
VectSp of K, W be Subspace of V for f be linear-Functional of V st the carrier
  of W c= ker f holds QFunctional(f,W) is homogeneous
proof
  let F be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
  let V be VectSp of F;
  let W be Subspace of V;
  let f be linear-Functional of V;
  set qf = QFunctional(f,W);
  set vq = VectQuot(V,W);
  assume
A1: the carrier of W c= ker f;
  now
    set C = CosetSet(V,W);
    let A be Vector of vq;
    let r be Scalar of vq;
A2: C = the carrier of vq by Def6;
    then A in C;
    then consider aa be Coset of W such that
A3: A = aa;
    consider a be Vector of V such that
A4: aa = a+W by VECTSP_4:def 6;
    r*A = (the lmult of vq).(r,A)
      .= (lmultCoset(V,W)).(r,A) by Def6
      .= r*a+ W by A2,A3,A4,Def5;
    hence qf.(r*A) = f.(r*a) by A1,Def12
      .= r*f.a by HAHNBAN1:def 8
      .= r*(qf.A) by A1,A3,A4,Def12;
  end;
  hence thesis;
end;
