
theorem Th63:
  for V,W be VectSp of F_Complex, f be sesquilinear-Form of V,W
  holds Q*Form(f) = RQ*Form(LQForm(f)) & Q*Form(f) = LQForm(RQ*Form(f))
proof
  set K = F_Complex;
  let V,W be VectSp of K, f be sesquilinear-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(RQ*Form(f)), vql = VectQuot(
  V,LR);
A1: dom Q*Form(f) =[:the carrier of vq, the carrier of wq:] by FUNCT_2:def 1;
A2: now
A3: R = RL by Th61;
    let x be object;
    assume x in dom Q*Form(f);
    then consider A,B be object such that
A4: A in the carrier of vq and
A5: B in the carrier of wq and
A6: x=[A,B] by ZFMISC_1:def 2;
    reconsider A as Vector of vq by A4;
    consider v be Vector of V such that
A7: A = v + L by VECTSP10:22;
    reconsider B as Vector of wq by A5;
    consider w be Vector of W such that
A8: B = w + R by VECTSP10:22;
    thus (Q*Form(f)).x = (Q*Form(f)).(A,B) by A6
      .= f.(v,w) by A7,A8,Def12
      .= (LQForm(f)).(A,w) by A7,BILINEAR:def 20
      .=(RQ*Form(LQForm(f))).(A,B) by A8,A3,Th59
      .=(RQ*Form(LQForm(f))).x by A6;
  end;
  dom RQ*Form (LQForm(f))= [:the carrier of vq, the carrier of wqr:] & the
  carrier of wqr = the carrier of wq by Th61,FUNCT_2:def 1;
  hence Q*Form(f) = RQ*Form(LQForm(f)) by A1,A2;
A9: now
A10: L = LR by Th62;
    let x be object;
    assume x in dom Q*Form(f);
    then consider A,B be object such that
A11: A in the carrier of vq and
A12: B in the carrier of wq and
A13: x=[A,B] by ZFMISC_1:def 2;
    reconsider A as Vector of vq by A11;
    consider v be Vector of V such that
A14: A = v + L by VECTSP10:22;
    reconsider B as Vector of wq by A12;
    consider w be Vector of W such that
A15: B = w + R by VECTSP10:22;
    thus (Q*Form(f)).x = (Q*Form(f)).(A,B) by A13
      .= f.(v,w) by A14,A15,Def12
      .= (RQ*Form(f)).(v,B) by A15,Th59
      .=(LQForm(RQ*Form(f))).(A,B) by A14,A10,BILINEAR:def 20
      .=(LQForm(RQ*Form(f))).x by A13;
  end;
  dom LQForm (RQ*Form(f))= [:the carrier of vql, the carrier of wq:] & the
  carrier of vql = the carrier of vq by Th62,FUNCT_2:def 1;
  hence thesis by A1,A9;
end;
