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

theorem Th43:
  for N being normal SubLoop of Q holds
  the carrier of N = 1.Q * N
proof
  let N be normal SubLoop of Q;
  A1: the carrier of N c= the carrier of Q by Def30;
  thus the carrier of N c= 1.Q * N
  proof
    let x be object;
    assume A2: x in the carrier of N;
    then reconsider x as Element of Q by A1;
    A3: (curry (the multF of Q)).x in Mlt (@ ([#] N)) by Th32,A2;
    reconsider h = (curry (the multF of Q)).x as Permutation of Q
      by Th30;
    h.(1.Q) = x * 1.Q by FUNCT_5:69;
    hence thesis by Def39,A3;
  end;
  let x be object;
  assume x in 1.Q * N;
  then A4:ex h be Permutation of Q st
  h in Mlt (@ [#] N) & x = h.(1.Q) by Def39;
  1.N = 1.Q by Def30;
  hence thesis by Th42,A4;
end;
