
theorem Th25:
  for V be VectSp of F_Complex, W be Subspace of V
  for f be antilinear-Functional of V st the carrier of W c= ker f*' holds
  QFunctional(f,W) is cmplxhomogeneous
proof
  let V be VectSp of F_Complex, W be Subspace of V, f be antilinear-Functional
  of V;
  assume
A1: the carrier of W c= ker f*';
  set vq = VectQuot(V,W);
  set qf = QFunctional(f,W);
A2: ker f*' = ker f by Th23;
  now
    set C = CosetSet(V,W);
    let A be Vector of vq;
    let r be Scalar of vq;
A3: C = the carrier of vq by VECTSP10:def 6;
    then A in C;
    then consider aa be Coset of W such that
A4: A = aa;
    consider a be Vector of V such that
A5: aa = a+W by VECTSP_4:def 6;
    r*A = (lmultCoset(V,W)).(r,A) by VECTSP10:def 6
      .= r*a+ W by A3,A4,A5,VECTSP10:def 5;
    hence qf.(r*A) = f.(r*a) by A1,A2,VECTSP10:def 12
      .= r*'*f.a by Def1
      .= r*'*(qf.A) by A1,A2,A4,A5,VECTSP10:def 12;
  end;
  hence thesis;
end;
