
theorem Th45:
  for K be add-associative right_zeroed right_complementable
Abelian associative well-unital distributive non empty doubleLoopStr for V be
  non empty ModuleStr over K, W be VectSp of K for f be additiveFAF
  homogeneousFAF Form of V,W holds leftker f = leftker (RQForm f)
proof
  let K be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
  let V be non empty ModuleStr over K, W be VectSp of K;
  let f be additiveFAF homogeneousFAF Form of V,W;
  set rf = RQForm(f), qw = VectQuot(W,RKer f);
  thus leftker f c= leftker (RQForm 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,Def21
        .= 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 Def21
      .= 0.K by A5;
  end;
  hence thesis by A4;
end;
