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

theorem Th52:
  y in x * lp (Cent Q) iff ex z st z in Cent Q & y = x * z
proof
  thus y in x * lp (Cent Q) implies ex z st z in Cent Q & y = x * z
  proof
    assume y in x * lp (Cent Q);
    then y in x * Cent Q by Th25;
    then consider h being Permutation of Q such that
    A1: h in Mlt (Cent Q) & h.x = y by Def39;
    consider z such that
    A2: z in Cent Q & for v holds h.v = v * z by Th51,A1;
    take z;
    thus thesis by A2,A1;
  end;
  given z such that
  A3: z in Cent Q & y = x * z;
  reconsider h = (curry' (the multF of Q)).(z) as Permutation of Q
  by Th31;
  ex h being Permutation of Q st h in Mlt (Cent Q) & h.x = y
  proof
    reconsider h = (curry' (the multF of Q)).(z) as Permutation of Q
    by Th31;
    take h;
    thus thesis by FUNCT_5:70,Th33,A3;
  end;
  then y in x * Cent Q by Def39;
  hence thesis by Th25;
end;
