reserve x,y for object;
reserve S for non void non empty ManySortedSign,
  o for OperSymbol of S,
  U0,U1, U2 for MSAlgebra over S;
reserve U0 for non-empty MSAlgebra over S;

theorem Th35:
  for S be non void non empty ManySortedSign, U0 be non-empty
  MSAlgebra over S, B be MSSubset of U0 st B = the Sorts of U0 holds Top (
  MSSubAlLattice(U0)) = GenMSAlg(B)
proof
  let S be non void non empty ManySortedSign, U0 be non-empty MSAlgebra over S
  , B be MSSubset of U0;
  reconsider G = GenMSAlg(B) as Element of MSSub(U0) by Def19;
  set L = MSSubAlLattice(U0);
  reconsider G1 = G as Element of L;
  assume
A1: B = the Sorts of U0;
  now
    let a be Element of L;
    reconsider a1 = a as Element of MSSub(U0);
    reconsider a2 = a1 as strict MSSubAlgebra of U0 by Def19;
    thus G1"\/"a = GenMSAlg(B)"\/"a2 by Def20
      .= G1 by A1,Th25;
    hence a "\/" G1 = G1;
  end;
  hence thesis by LATTICES:def 17;
end;
