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

theorem Th23:
  x in Nucl Q & y in Nucl Q implies x / y in Nucl Q
proof
  assume that
  A1: x in Nucl Q
  and
  A2: y in Nucl Q;
  A3: x in Nucl_l Q by Th12,A1;
  A4: x in Nucl_m Q by Th12,A1;
  A5: x in Nucl_r Q by Th12,A1;
  A6: y in Nucl_l Q by Th12,A2;
  A7: y in Nucl_m Q by Th12,A2;
  A8: y in Nucl_r Q by Th12,A2;
  for z,w holds ((x / y) * z) * w = (x / y) * (z * w)
  proof
    let z,w;
    ((x / y) * z) * w = ((x / y) * (y * (y \ z))) * w
    .= (((x / y) * y) * (y \ z)) * w by A7,Def23
    .= ((x / y) * y) * ((y \ z) * w) by A3,Def22
    .= (x / y) * (y * ((y \ z) * w)) by A7,Def23
    .= (x / y) * ((y * (y \ z)) * w) by A6,Def22
    .= (x / y) * (z * w);
    hence thesis;
  end;
  then A9: x / y in Nucl_l Q by Def22;
  for z,w holds (z * (x / y)) * w = z * ((x / y) * w)
  proof
    let z,w;
    (z * (x / y)) * w = (z * (x / y)) * (y * (y \ w))
    .= ((z * (x / y)) * y) * (y \ w) by A7,Def23
    .= (z * ((x / y) * y)) * (y \ w) by A8,Def24
    .= z * (((x / y) * y) * (y \ w)) by A4, Def23
    .= z * ((x / y) * (y * (y \ w))) by A7,Def23
    .= z * ((x / y) * w);
    hence thesis;
  end;
  then A10: x / y in Nucl_m Q by Def23;
  for z,w holds (z * w) * (x / y) = z * (w * (x / y))
  proof
    let z,w;
    ((z * w) * (x / y)) * y = (z * w) * ((x / y) * y) by A8,Def24
    .= z * (w * ((x / y) * y)) by A5,Def24
    .= z * ((w * (x / y)) * y) by A8,Def24
    .= (z * (w * (x / y))) * y by A8,Def24;
    hence thesis by Th2;
  end;
  then x / y in Nucl_r Q by Def24;
  hence thesis by Th12,A9,A10;
end;
