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

theorem
  for Q2 being multLoop holds
  for f being homomorphic Function of Q,Q2 holds
  lp (Ker f) is normal
proof
  let Q2 be multLoop;
  let f be homomorphic Function of Q,Q2;
  set H = lp (Ker f);
  A1: for x,y holds (x * H) * (y * H) = (x * y) * H
  proof
    let x,y;
    for z holds z in (x * H) * (y * H) iff z in (x * y) * H
    proof
      let z;
      thus z in (x * H) * (y * H) implies z in (x * y) * H
      proof
        assume z in (x * H) * (y * H);
        then consider v,w such that
        A2: v in x * H & w in y * H & z = v * w by Def42;
        A3: f.y = f.w by Th47,A2;
        f.z = f.v * f.w by Def28b,A2
        .= f.x * f.y by Th47,A2,A3
        .= f.(x * y) by Def28b;
        hence z in (x * y) * H by Th47;
      end;
      assume z in (x * y) * H;
      then A4: f.z = f.(x * y) by Th47;
      ex v,w st v in x * H & w in y * H & z = v * w
      proof
        take z / y,y;
        A5: f.(z / y) = f.z / f.y by Th14
        .= (f.x * f.y) / f.y by A4,Def28b
        .= f.x;
        f.y = f.y;
        hence thesis by A5,Th47;
      end;
      hence z in (x * H) * (y * H) by Def42;
    end;
    hence thesis by SUBSET_1:3;
  end;
  for x,y holds
  (x * H) * (y * H) = (x * y) * H
  &
  for z holds
  ((x * H) * (y * H) = (x * H) * (z * H) implies (y * H) = (z * H))
  &
  ((y * H) * (x * H) = (z * H) * (x * H) implies (y * H) = (z * H))
  proof
    let x,y;
    thus (x * H) * (y * H) = (x * y) * H by A1;
    let z;
    thus (x * H) * (y * H) = (x * H) * (z * H) implies (y * H) = (z * H)
    proof
      assume (x * H) * (y * H) = (x * H) * (z * H);
      then (x * y) * H = (x * H) * (z * H) by A1;
      then A6: (x * y) * H = (x * z) * H by A1;
      f.y = f.x \ (f.x * f.y)
      .= f.x \ f.(x * y) by Def28b
      .= f.x \ f.(x * z) by A6,Th48
      .= f.x \ (f.x * f.z) by Def28b
      .= f.z;
      hence (y * H) = (z * H) by Th48;
    end;
    assume (y * H) * (x * H) = (z * H) * (x * H);
    then (y * x) * H = (z * H) * (x * H) by A1;
    then A7: (y * x) * H = (z * x) * H by A1;
    f.y = (f.y * f.x) / f.x
    .= f.(y * x) / f.x by Def28b
    .= f.(z * x) / f.x by A7,Th48
    .= (f.z * f.x) / f.x by Def28b
    .= f.z;
    hence (y * H) = (z * H) by Th48;
  end;
  hence thesis;
end;
