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

theorem Th42:
  for N being SubLoop of Q holds
  for f being Function of Q,Q st f in Mlt (@ ([#] N)) holds
  for x holds x in (@ ([#] N)) iff f.x in (@ ([#] N))
proof
  let N be SubLoop of Q;
  reconsider H = @ ([#] N) as Subset of Q;
  defpred P[Function of Q,Q] means for x holds x in H iff $1.x in H;
  A1: for u being Element of Q st u in H holds
  for f being Function of Q,Q st for x being Element of Q holds
    f.x = x * u holds P[f]
  proof
    let u;
    assume A2:u in H;
    let f be Function of Q,Q;
    assume A3: for x holds f.x = x * u;
    P[f]
    proof
      let x;
      thus x in H implies f.x in H
      proof
        assume x in H;
        then x * u in the carrier of N by Th37,A2;
        hence thesis by A3;
      end;
      assume f.x in H;
      then reconsider xu1 = x * u as Element of N by A3;
      reconsider u1 = u as Element of N by A2;
      the carrier of N c= the carrier of Q by Def30;
      then reconsider xu1u1 = xu1 / u1 as Element of Q;
      A4: x * u = (xu1 / u1) * u1
      .= xu1u1 * u by Th36;
      x = (xu1u1 * u) / u by A4
      .= (xu1 / u1);
      hence thesis;
    end;
    hence thesis;
  end;
  A5: for u being Element of Q st u in H holds
  for f being Function of Q,Q st for x being Element of Q holds
    f.x = u * x holds P[f]
  proof
    let u;
    assume A6: u in H;
    let f be Function of Q,Q;
    assume A7: for x holds f.x = u * x;
    P[f]
    proof
      let x;
      thus x in H implies f.x in H
      proof
        assume x in H;
        then u * x in the carrier of N by Th37,A6;
        hence thesis by A7;
      end;
      assume f.x in H;
      then reconsider ux1 = u * x,u1=u as Element of N by A7,A6;
      the carrier of N c= the carrier of Q by Def30;
      then reconsider u1ux1 = u1 \ ux1 as Element of Q;
      u * x = u1 * (u1 \ ux1)
      .= u * u1ux1 by Th36;
      then x  = u \ (u * u1ux1)
      .= (u1 \ ux1);
      hence thesis;
    end;
    hence thesis;
  end;
  A8: for g,h being Permutation of Q st P[g] & P[h] holds P[g*h]
  proof
    let g,h be Permutation of Q such that A9: P[g] & P[h];
    let x;
    x in H iff h.x in H by A9;
    then x in H iff g.(h.x) in H by A9;
    hence thesis by FUNCT_2:15;
  end;
  A10: for g being Permutation of Q st P[g] holds P[g"]
  proof
    let g be Permutation of Q such that A11: P[g];
    let x;
    x = (id the carrier of Q).x
    .= (g*(g")).x by FUNCT_2:61
    .= g.((g").x) by FUNCT_2:15;
    hence x in H iff (g").x in H by A11;
  end;
  for f being Function of Q,Q st f in Mlt H holds P[f]
    from MltInd(A1,A5,A8,A10);
  hence thesis;
end;
