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

theorem Th25:
  [#]lp (Cent Q) = Cent Q
proof
  A1: 1.Q in Cent Q
  proof
    A2: 1.Q in Nucl Q by Th20;
    for y holds 1.Q * y = y * 1.Q;
    then 1.Q in Comm Q by Def25;
    hence thesis by XBOOLE_0:def 4, A2;
  end;
  A3: for x,y st x in Cent Q & y in Cent Q holds x * y in Cent Q
  proof
    let x,y;
    assume that
    A4: x in Cent Q
    and
    A5: y in Cent Q;
    A6: x in Comm Q & x in Nucl Q by XBOOLE_0:def 4, A4;
    A7: y in Comm Q & y in Nucl Q by XBOOLE_0:def 4, A5;
    A8: x in Nucl_l Q by Th12,A6;
    A9: y in Nucl_m Q & y in Nucl_r Q by Th12,A7;
    for z holds (x * y) * z = z * (x * y)
    proof
      let z;
      (x * y) * z = x * (y * z) by A9,Def23
      .= x * (z * y) by A7,Def25
      .= (x * z) * y by A8,Def22
      .= (z * x) * y by A6,Def25
      .= z * (x * y) by A9,Def24;
      hence thesis;
    end;
    then A10: x * y in Comm Q by Def25;
    x * y in Nucl Q by Th21,A6,A7;
    hence x * y in Cent Q by XBOOLE_0:def 4,A10;
  end;
  A11: for x,y st x in Cent Q & y in Cent Q holds x \ y in Cent Q
  proof
    let x,y;
    assume that
    A12: x in Cent Q
    and
    A13: y in Cent Q;
    A14: x in Comm Q & x in Nucl Q by XBOOLE_0:def 4, A12;
    A15: y in Comm Q & y in Nucl Q by XBOOLE_0:def 4, A13;
    A16: x in Nucl_m Q by Th12,A14;
    for z holds (x \ y) * z = z * (x \ y)
    proof
      let z;
      (x \ y) * z = (x \ y) * ((z / x) * x)
      .= (x \ y) * (x * (z / x)) by A14,Def25
      .= ((x \ y) * x) * (z / x) by A16,Def23
      .= (x * (x \ y)) * (z / x) by A14,Def25
      .= (z / x) * (x * (x \ y)) by A15,Def25
      .= ((z / x) * x) * (x \ y) by A16,Def23
      .= z * (x \ y);
      hence thesis;
    end;
    then A17: x \ y in Comm Q by Def25;
    x \ y in Nucl Q by Th22,A14,A15;
    hence x \ y in Cent Q by XBOOLE_0:def 4, A17;
  end;
  for x,y st x in Cent Q & y in Cent Q holds x / y in Cent Q
  proof
    let x,y;
    assume that
    A18: x in Cent Q
    and
    A19: y in Cent Q;
    A20: x in Comm Q & x in Nucl Q by XBOOLE_0:def 4, A18;
    A21: y in Comm Q & y in Nucl Q by XBOOLE_0:def 4, A19;
    A22: y in Nucl_m Q by Th12,A21;
    for z holds (x / y) * z = z * (x / y)
    proof
      let z;
      thus (x / y) * z = (x / y) * ((z / y) * y)
      .= (x / y) * (y * (z / y)) by A21,Def25
      .= ((x / y) * y) * (z / y) by A22,Def23
      .= (z / y) * ((x / y) * y) by A20,Def25
      .= (z / y) * (y * (x / y)) by A21,Def25
      .= ((z / y) * y) * (x / y) by A22,Def23
      .= z * (x / y);
    end;
    then A23: x / y in Comm Q by Def25;
    x / y in Nucl Q by Th23,A20,A21;
    hence x / y in Cent Q by XBOOLE_0:def 4,A23;
  end;
  hence thesis by Th18,A1,A3,A11;
end;
