
theorem Th55:
  for K be add-associative right_zeroed right_complementable
commutative Abelian associative well-unital distributive almost_left_invertible
non empty doubleLoopStr for V be non empty ModuleStr over K, W be VectSp of K
  for f be Functional of V, g be linear-Functional of W st f <> 0Functional(V)
  holds RKer FormFunctional(f,g) = Ker g & RQForm(FormFunctional(f,g)) =
  FormFunctional(f,CQFunctional(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 non empty ModuleStr over K, W be VectSp of K, f be
  Functional of V, g be linear-Functional of W;
  set fg = FormFunctional(f,g), cg = CQFunctional(g), fcfg = FormFunctional(f,
  CQFunctional(g)), wqr = VectQuot(W, RKer fg), wq =VectQuot(W, Ker g);
  assume f <> 0Functional(V);
  then
A1: rightker fg = ker g by Th53;
  the carrier of RKer fg = rightker fg by Def19;
  hence
A2: RKer fg = Ker g by A1,VECTSP10:def 11;
A3: now
    let x be object;
    assume x in dom fcfg;
    then consider A be Vector of V, B be Vector of wq such that
A4: x=[A,B] by DOMAIN_1:1;
    consider w be Vector of W such that
A5: B = w + Ker g by VECTSP10:22;
    thus fcfg.x = fcfg.(A,B) by A4
      .= f.A * cg.B by Def10
      .=f.A *g.w by A5,VECTSP10:35
      .= fg.(A,w) by Def10
      .= (RQForm(fg)).(A,B) by A2,A5,Def21
      .= (RQForm(fg)).x by A4;
  end;
  dom RQForm(fg) = [: the carrier of V, the carrier of wqr:] & dom fcfg =
  [: the carrier of V, the carrier of wq:] by FUNCT_2:def 1;
  hence thesis by A2,A3,FUNCT_1:2;
end;
