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

theorem Th44:
  for N being normal SubLoop of Q holds
  Ker (QuotientHom(Q,N)) = @ ([#] N)
proof
  let N be normal SubLoop of Q;
  A1: the carrier of N c= the carrier of Q by Def30;
  set f = QuotientHom(Q,N);
  for x holds x in Ker f iff x in @ ([#] N)
  proof
    let x;
    thus x in Ker f implies x in @ ([#] N)
    proof
      assume A2: x in Ker f;
      A3:x * N = f.x by Def48
      .= 1.(Q _/_ N) by Def29,A2
      .= 1.Q * N;
      A4: 1.N = 1.Q by Def30;
      reconsider h = (curry (the multF of Q)).(1.Q) as Permutation of Q
        by Th30;
      A5: h in Mlt (@ [#] N) by A4,Th32;
      A6: h.x in x * (@ ([#] N)) by Def39,A5;
      h.x = 1.Q * x by FUNCT_5:69;
      hence thesis by A6,A3,Th43;
    end;
    assume A7: x in @ ([#] N);
    A8: for y holds y in x * N iff y in 1.Q * N
    proof
      let y;
      thus y in x * N implies y in 1.Q * N
      proof
        assume y in x * N;
        then consider h being Permutation of Q such that
        A9: h in Mlt (@ ([#] N)) & h.x = y by Def39;
        h.x in @ ([#] N) by Th42,A9,A7;
        hence thesis by A9,Th43;
      end;
      assume y in 1.Q * N;
      then reconsider y1 = y as Element of N by Th43;
      reconsider x1 = x as Element of N by A7;
      ex h being Permutation of Q st h in Mlt (@ ([#] N)) & y = h.x
      proof
        reconsider y1x1 = y1 / x1 as Element of Q by A1;
        reconsider h = (curry (the multF of Q)).(y1x1) as Permutation of Q
        by Th30;
        take h;
        thus h in Mlt (@ [#] N) by Th32;
        h.x = y1x1 * x by FUNCT_5:69
        .= (y / x) * x by Th40
        .= y;
        hence h.x = y;
      end;
      hence thesis by Def39;
    end;
    f.x = x * N by Def48
    .= 1.(Q _/_ N) by A8,SUBSET_1:3;
    hence thesis by Def29;
  end;
  hence thesis by SUBSET_1:3;
end;
