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

theorem Th18:
  for H being Subset of Q st
  1.Q in H & (for x,y st x in H & y in H holds x * y in H)
  & (for x,y st x in H & y in H holds x \ y in H)
  & (for x,y st x in H & y in H holds x / y in H)
  holds
  [#]lp H = H
proof
  let H be Subset of Q;
  assume that
  A1: 1.Q in H
  and
  A2: for x,y st x in H & y in H holds x * y in H
  and
  A3: for x,y st x in H & y in H holds x \ y in H
  and
  A4: for x,y st x in H & y in H holds x / y in H;
  reconsider ONE=1.Q as Element of H by A1;
  set mm = (the multF of Q)||H;
  now let x be set such that A5: x in [: H,H:];
    consider x1,x2 be object such that
    A6:x1 in H & x2 in H & x=[x1,x2] by A5,ZFMISC_1:def 2;
    reconsider x1,x2 as Element of Q by A6;
    x1*x2 in H by A6,A2;
    hence (the multF of Q).x in H by A6;
  end;
  then
  H is Preserv of the multF of Q by REALSET1:def 1;
  then reconsider mm as BinOp of H by REALSET1:2;
  set LP = multLoopStr(#H,mm,ONE#);
  reconsider LP as non empty SubLoopStr of Q by A1,Def30;
  LP is SubLoop of Q
  proof
    now
      let x be Element of LP;
      x in the carrier of LP;
      then reconsider x1=x as Element of Q;
      x*1.LP = x1*1.Q & 1.LP*x = 1.Q*x1 by RING_3:1;
      hence x * 1. LP = x & 1.LP * x = x;
    end;
    then  A7: LP is well-unital;
    A8: LP is invertible
    proof
      hereby let x,y be Element of LP;
        x in the carrier of LP &   y in the carrier of LP;
        then reconsider x1=x,y1=y as Element of Q;
        reconsider z=x1 \ y1 as Element of LP by A3;
        take z;
        thus x*z = x1 * (x1 \ y1) by RING_3:1
        .= y;
      end;
      hereby let x,y be Element of LP;
        x in the carrier of LP &   y in the carrier of LP;
        then reconsider x1=x,y1=y as Element of Q;
        reconsider z=y1 / x1 as Element of LP by A4;
        take z;
        thus z*x = (y1 / x1) * x1 by RING_3:1
        .= y;
      end;
    end;
    LP is  cancelable
    proof
      thus LP is left_mult-cancelable
      proof
        let x be Element of LP;
        let y,z be Element of LP;
        x in the carrier of LP
        & y in the carrier of LP
        & z in the carrier of LP;
        then reconsider x1=x,y1=y,z1=z as Element of Q;
        x1*y1 = x*y & x1*z1 = x*z by RING_3:1;
        hence thesis by ALGSTR_0:def 20;
      end;
      let x be Element of LP;
      let y,z be Element of LP;
      x in the carrier of LP & y in the carrier of LP & z in the carrier of LP;
      then reconsider x1=x,y1=y,z1=z as Element of Q;
      y1*x1 = y*x & z1*x1 = z*x by RING_3:1;
      hence thesis by ALGSTR_0:def 21;
    end;
    hence thesis by A7,A8;
  end;
  then reconsider LP as strict SubLoop of Q;
  [#](lp H) c= [#]LP = H by Def33;
  hence thesis by Def33;
end;
