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
  for S be non void non empty ManySortedSign, U0 be non-empty MSAlgebra
  over S holds Bottom (MSSubAlLattice(U0)) = GenMSAlg(Constants(U0))
proof
  let S be non void non empty ManySortedSign, U0 be non-empty MSAlgebra over S;
  set C = Constants(U0);
  reconsider G = GenMSAlg(C) as Element of MSSub(U0) by Def19;
  set L = MSSubAlLattice(U0);
  reconsider G1 = G as Element of L;
  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(C) /\ a2 by Def21
      .= G1 by Th23;
    hence a "/\" G1 = G1;
  end;
  hence thesis by LATTICES:def 16;
end;
