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

theorem Th22:
  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 * (((x \ y) * z) * w) = (x * ((x \ y) * z)) * w by A3,Def22
    .= ((x * (x \ y)) * z) * w by A3,Def22
    .= (x * (x \ y)) * (z * w) by A6,Def22
    .= x * ((x \ y) * (z * w)) by A3,Def22;
    hence thesis by Th1;
  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) * x) * (x \ y)) * w
    .= ((z / x) * (x * (x \ y))) * w by A4,Def23
    .= (z / x) * ((x * (x \ y)) * w) by A7,Def23
    .= (z / x) * (x * ((x \ y) * w)) by A3,Def22
    .= ((z / x) * x) * ((x \ y) * w) by A4,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) = (z * ((w / x) * x)) * (x \ y)
    .= ((z * (w / x)) * x) * (x \ y) by A5,Def24
    .= (z * (w / x)) * (x * (x \ y)) by A4,Def23
    .= z * ((w / x) * (x * (x \ y)))by A8,Def24
    .= z * (((w / x) * x) * (x \ y)) by A4,Def23
    .= z * (w * (x \ y));
    hence thesis;
  end;
  then x \ y in Nucl_r Q by Def24;
  hence thesis by Th12,A9,A10;
end;
