reserve S for OrderSortedSign;

theorem
  for S be OrderSortedSign, U0 be strict non-empty OSAlgebra of S, A be
  MSSubset of U0 holds A is OSGeneratorSet of U0 iff for O being OSSubset of U0
  st O = OSCl A holds GenOSAlg(O) = U0
proof
  let S be OrderSortedSign, U0 be strict non-empty OSAlgebra of S, A be
  MSSubset of U0;
  thus A is OSGeneratorSet of U0 implies for O being OSSubset of U0 st O =
  OSCl A holds GenOSAlg(O) = U0
  proof
    reconsider U1 = U0 as MSSubAlgebra of U0 by MSUALG_2:5;
    assume
A1: A is OSGeneratorSet of U0;
    let O be OSSubset of U0;
    assume O = OSCl A;
    then the Sorts of GenOSAlg(O) = the Sorts of U1 by A1,Def1;
    hence thesis by MSUALG_2:9;
  end;
  assume
A2: for O being OSSubset of U0 st O = OSCl A holds GenOSAlg(O) = U0;
  let O be OSSubset of U0;
  assume O = OSCl A;
  hence thesis by A2;
end;
