
theorem Th60:
  for V be non empty ModuleStr over F_Complex, W be VectSp of
F_Complex for f be additiveFAF cmplxhomogeneousFAF Form of V,W holds leftker f
  = leftker (RQ*Form f)
proof
  set K = F_Complex;
  let V be non empty ModuleStr over K, W be VectSp of K;
  let f be additiveFAF cmplxhomogeneousFAF Form of V,W;
  set rf = RQ*Form(f), qw = VectQuot(W,RKer f*');
  thus leftker f c= leftker (RQ*Form f)
  proof
    let x be object;
    assume x in leftker f;
    then consider v be Vector of V such that
A1: x=v and
A2: for w be Vector of W holds f.(v,w) = 0.K;
    now
      let A be Vector of qw;
      consider w be Vector of W such that
A3:   A = w+RKer (f*') by VECTSP10:22;
      thus rf.(v,A) = f.(v,w) by A3,Th59
        .= 0.K by A2;
    end;
    hence thesis by A1;
  end;
  let x be object;
  assume x in leftker rf;
  then consider v be Vector of V such that
A4: x=v and
A5: for A be Vector of qw holds rf.(v,A) = 0.K;
  now
    let w be Vector of W;
    reconsider A = w + RKer f*' as Vector of qw by VECTSP10:23;
    thus f.(v,w) = rf.(v,A) by Th59
      .= 0.K by A5;
  end;
  hence thesis by A4;
end;
