
theorem Th44:
  for K be add-associative right_zeroed right_complementable
Abelian associative well-unital distributive non empty doubleLoopStr for V be
  VectSp of K, W be non empty ModuleStr over K for f be additiveSAF
  homogeneousSAF Form of V,W holds rightker f = rightker (LQForm f)
proof
  let K be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
  let V be VectSp of K, W be non empty ModuleStr over K;
  let f be additiveSAF homogeneousSAF Form of V,W;
  set lf = LQForm(f), qv = VectQuot(V,LKer f);
  thus rightker f c= rightker (LQForm f)
  proof
    let x be object;
    assume x in rightker f;
    then consider w be Vector of W such that
A1: x=w and
A2: for v be Vector of V holds f.(v,w) = 0.K;
    now
      let A be Vector of qv;
      consider v be Vector of V such that
A3:   A = v+LKer f by VECTSP10:22;
      thus lf.(A,w) = f.(v,w) by A3,Def20
        .= 0.K by A2;
    end;
    hence thesis by A1;
  end;
  let x be object;
  assume x in rightker lf;
  then consider w be Vector of W such that
A4: x=w and
A5: for A be Vector of qv holds lf.(A,w) = 0.K;
  now
    let v be Vector of V;
    reconsider A = v + LKer f as Vector of qv by VECTSP10:23;
    thus f.(v,w) = lf.(A,w) by Def20
      .= 0.K by A5;
  end;
  hence thesis by A4;
end;
