reserve x,y for set,
  n for Nat;

theorem
  for U0 be strict Universal_Algebra,A be Subset of U0 st Constants U0
  <> {} or A <> {} holds A is GeneratorSet of U0 iff GenUnivAlg(A) = U0
proof
  let U0 be strict Universal_Algebra,A be Subset of U0 such that
A1: Constants U0 <> {} or A <> {};
  thus A is GeneratorSet of U0 implies GenUnivAlg(A) = U0
  proof
    assume A is GeneratorSet of U0;
    then
A2: the carrier of GenUnivAlg(A) = the carrier of U0 by A1,Lm3;
    U0 is strict SubAlgebra of U0 by UNIALG_2:8;
    hence thesis by A2,UNIALG_2:12;
  end;
  assume GenUnivAlg(A) = U0;
  hence thesis by A1,Lm3;
end;
