
theorem Th54:
  for K be add-associative right_zeroed right_complementable
commutative Abelian associative well-unital distributive almost_left_invertible
non empty doubleLoopStr for V be VectSp of K, W be non empty ModuleStr over K
  for f be linear-Functional of V, g be Functional of W st g <> 0Functional(W)
  holds LKer FormFunctional(f,g) = Ker f & LQForm(FormFunctional(f,g)) =
  FormFunctional(CQFunctional(f),g)
proof
  let K be add-associative right_zeroed right_complementable commutative
Abelian associative well-unital distributive almost_left_invertible non empty
  doubleLoopStr, V be VectSp of K, W be non empty ModuleStr over K, f be
  linear-Functional of V, g be Functional of W;
  set fg = FormFunctional(f,g), cf = CQFunctional(f), fcfg = FormFunctional(
  CQFunctional(f),g), vql = VectQuot(V, LKer fg), vq =VectQuot(V, Ker f);
  assume g <> 0Functional(W);
  then
A1: leftker fg = ker f by Th51;
  the carrier of LKer fg = leftker fg by Def18;
  hence
A2: LKer fg = Ker f by A1,VECTSP10:def 11;
A3: now
    let x be object;
    assume x in dom fcfg;
    then consider A be Vector of vq, B be Vector of W such that
A4: x=[A,B] by DOMAIN_1:1;
    consider v be Vector of V such that
A5: A = v + Ker f by VECTSP10:22;
    thus fcfg.x = fcfg.(A,B) by A4
      .= cf.A * g.B by Def10
      .=f.v *g.B by A5,VECTSP10:35
      .= fg.(v,B) by Def10
      .= (LQForm(fg)).(A,B) by A2,A5,Def20
      .= (LQForm(fg)).x by A4;
  end;
  dom LQForm(fg) = [: the carrier of vql, the carrier of W:] & dom fcfg =
  [: the carrier of vq, the carrier of W:] by FUNCT_2:def 1;
  hence thesis by A2,A3,FUNCT_1:2;
end;
