reserve Q,Q1,Q2 for multLoop;
reserve x,y,z,w,u,v for Element of Q;

theorem Th19:
  for f being homomorphic Function of Q,Q2 holds
  [#]lp (Ker f) = Ker f
  proof
    let f be homomorphic Function of Q,Q2;
    f.(1.Q) = 1.Q2 by Def28a;
    then A1: 1.Q in Ker f by Def29;
    A2: for x,y st x in Ker f & y in Ker f holds x * y in Ker f
    proof
      let x,y be Element of Q;
      assume that
      A3: x in Ker f
      and
      A4: y in Ker f;
      f.(x * y) = f.x * f.y by Def28b
      .= 1.Q2 * f.y by Def29,A3
      .= 1.Q2 by Def29,A4;
      hence x*y in Ker f by Def29;
    end;
    A5: for x,y st x in Ker f & y in Ker f holds x \ y in Ker f
    proof
      let x,y be Element of Q;
      assume that
      A6: x in Ker f
      and
      A7: y in Ker f;
      f.(x \ y) = f.x \ f.y by Th13
      .= 1.Q2 \ f.y by Def29,A6
      .= 1.Q2 by Def29,A7;
      hence x\y in Ker f by Def29;
    end;
    for x,y st x in Ker f & y in Ker f holds x / y in Ker f
    proof
      let x,y be Element of Q;
      assume that
      A8: x in Ker f
      and
      A9: y in Ker f;
      f.(x / y) = f.x / f.y by Th14
      .= f.x / 1.Q2 by Def29,A9
      .= 1.Q2 by Def29,A8;
      hence x/y in Ker f by Def29;
    end;
    hence thesis by Th18,A1,A2,A5;
  end;
