reserve x for set,
  R for non empty Poset;
reserve S1 for OrderSortedSign,
  OU0 for OSAlgebra of S1;
reserve s,s1,s2,s3,s4 for SortSymbol of S1;
reserve U0 for non-empty OSAlgebra of S1;

theorem
  for U0 be non-empty OSAlgebra of S1 holds Bottom (OSSubAlLattice(U0))
  = GenOSAlg(OSConstants(U0))
proof
  let U0 be non-empty OSAlgebra of S1;
  set C = OSConstants(U0);
  reconsider G = GenOSAlg(C) as Element of OSSub(U0) by Def14;
  set L = OSSubAlLattice(U0);
  reconsider G1 = G as Element of L;
  now
    let a be Element of L;
    reconsider a1 = a as Element of OSSub(U0);
    reconsider a2 = a1 as strict OSSubAlgebra of U0 by Def14;
    thus G1 "/\" a = GenOSAlg(C) /\ a2 by Def16
      .= G1 by Th36;
    hence a "/\" G1 = G1;
  end;
  hence thesis by LATTICES:def 16;
end;
