
theorem Th48:
  for K be add-associative right_zeroed right_complementable
Abelian associative well-unital distributive non empty doubleLoopStr for V,W
be VectSp of K, f be bilinear-Form of V,W holds QForm(f) = RQForm(LQForm(f)) &
  QForm(f) = LQForm(RQForm(f))
proof
  let K be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
  let V,W be VectSp of K, f be bilinear-Form of V,W;
  set L = LKer(f), vq = VectQuot(V,L), R = RKer(f), wq = VectQuot(W,R), RL =
RKer(LQForm(f)), wqr = VectQuot(W,RL), LR = LKer(RQForm(f)), vql = VectQuot(V,
  LR);
A1: dom QForm(f) = [:the carrier of vq, the carrier of wq:] by FUNCT_2:def 1;
A2: now
    let x be object;
    assume x in dom QForm(f);
    then consider A be Vector of vq, B be Vector of wq such that
A3: x=[A,B] by DOMAIN_1:1;
    consider w be Vector of W such that
A4: B = w + R by VECTSP10:22;
A5: R = RL by Th46;
    consider v be Vector of V such that
A6: A = v + L by VECTSP10:22;
    thus (QForm(f)).x = (QForm(f)).(A,B) by A3
      .= f.(v,w) by A6,A4,Def22
      .= (LQForm(f)).(A,w) by A6,Def20
      .=(RQForm(LQForm(f))).(A,B) by A4,A5,Def21
      .=(RQForm(LQForm(f))).x by A3;
  end;
  dom RQForm (LQForm(f))= [:the carrier of vq, the carrier of wqr:] & the
  carrier of wqr = the carrier of wq by Th46,FUNCT_2:def 1;
  hence QForm(f) = RQForm(LQForm(f)) by A1,A2,FUNCT_1:2;
A7: now
    let x be object;
    assume x in dom QForm(f);
    then consider A be Vector of vq, B be Vector of wq such that
A8: x=[A,B] by DOMAIN_1:1;
    consider w be Vector of W such that
A9: B = w + R by VECTSP10:22;
A10: L = LR by Th47;
    consider v be Vector of V such that
A11: A = v + L by VECTSP10:22;
    thus (QForm(f)).x = (QForm(f)).(A,B) by A8
      .= f.(v,w) by A11,A9,Def22
      .= (RQForm(f)).(v,B) by A9,Def21
      .=(LQForm(RQForm(f))).(A,B) by A11,A10,Def20
      .=(LQForm(RQForm(f))).x by A8;
  end;
  dom LQForm (RQForm(f))= [:the carrier of vql, the carrier of wq:] & the
  carrier of vql = the carrier of vq by Th47,FUNCT_2:def 1;
  hence thesis by A1,A7,FUNCT_1:2;
end;
