reserve S for non void non empty ManySortedSign,
  U0 for MSAlgebra over S;

theorem
  for S be non void non empty ManySortedSign, U0 be strict non-empty
  MSAlgebra over S, A be MSSubset of U0 holds A is GeneratorSet of U0 iff
  GenMSAlg(A) = U0
proof
  let S be non void non empty ManySortedSign, U0 be strict non-empty MSAlgebra
  over S, A be MSSubset of U0;
  thus A is GeneratorSet of U0 implies GenMSAlg(A) = U0
  proof
    reconsider U1 = U0 as MSSubAlgebra of U0 by MSUALG_2:5;
    assume A is GeneratorSet of U0;
    then the Sorts of GenMSAlg(A) = the Sorts of U1 by Def4;
    hence thesis by MSUALG_2:9;
  end;
  assume GenMSAlg(A) = U0;
  hence thesis by Def4;
end;
