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

theorem Th53:
  x * lp (Cent Q) = y * lp (Cent Q) iff ex z st z in Cent Q & y = x * z
proof
  thus x * lp (Cent Q) = y * lp (Cent Q) implies
    ex z st z in Cent Q & y = x * z
  proof
    assume A1: x * lp (Cent Q) = y * lp (Cent Q);
    1.Q in Cent Q & y = y * 1.Q by Th50;
    hence ex z st z in Cent Q & y = x * z by A1,Th52;
  end;
  thus (ex z st z in Cent Q & y = x * z) implies
    x * lp (Cent Q) = y * lp (Cent Q)
  proof
    assume ex z st z in Cent Q & y = x * z;
    then consider z such that
    A2: z in Cent Q & y = x * z;
    z in Nucl Q by A2,XBOOLE_0:def 4;
    then A3: z in Nucl_m Q by Th12;
    for w holds w in x * lp (Cent Q) iff w in y * lp (Cent Q)
    proof
      let w;
      thus w in x * lp (Cent Q) implies w in y * lp (Cent Q)
      proof
        assume w in x * lp (Cent Q);
        then consider v such that
        A4: v in Cent Q & w = x * v by Th52;
        ex u st u in Cent Q & w = y * u
        proof
          take (z \ v);
          z in [#] (lp (Cent Q)) & v in [#] (lp (Cent Q)) by A2,A4,Th25;
          then z \ v in [#] (lp (Cent Q)) by Th39;
          hence (z \ v) in Cent Q by Th25;
          w = x * (z * (z \ v)) by A4
          .= y * (z \ v) by A2,Def23,A3;
          hence thesis;
        end;
        hence thesis by Th52;
      end;
      assume w in y * lp (Cent Q);
      then consider v such that
      A5: v in Cent Q & w = y * v by Th52;
      ex u st u in Cent Q & w = x * u
      proof
        take (z * v);
        z in [#] (lp (Cent Q)) & v in [#] (lp (Cent Q)) by A2,A5,Th25;
        then z * v in [#] (lp (Cent Q)) by Th37;
        hence thesis by Def23,A3,A2,A5,Th25;
      end;
      hence thesis by Th52;
    end;
    hence x * lp (Cent Q) = y * lp (Cent Q) by SUBSET_1:3;
  end;
end;
