reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th41:
  for U1,U2 being Universal_Algebra st the UAStr of U1 = the UAStr of U2
  for G being GeneratorSet of U1 holds G is GeneratorSet of U2
  proof
    let U1,U2 be Universal_Algebra;
    assume A1: the UAStr of U1 = the UAStr of U2;
    let G be GeneratorSet of U1;
    reconsider G2 = G as Subset of U2 by A1;
    G2 is GeneratorSet of U2
    proof
      let A be Subset of U2;
      reconsider B = A as Subset of U1 by A1;
      assume A is opers_closed;
      hence thesis by A1,Th40,FREEALG:def 4;
    end;
    hence G is GeneratorSet of U2;
  end;
