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 Th48:
  for U0 be non-empty OSAlgebra of S1 holds OSSubAlLattice(U0) is bounded
proof
  let U0 be non-empty OSAlgebra of S1;
  set L = OSSubAlLattice(U0);
  thus L is lower-bounded
  proof
    set C = OSConstants(U0);
    reconsider G = GenOSAlg(C) as Element of OSSub(U0) by Def14;
    reconsider G1 = G as Element of L;
    take G1;
    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 thesis;
  end;
  thus L is upper-bounded
  proof
    reconsider B = the Sorts of U0 as MSSubset of U0 by PBOOLE:def 18;
    the Sorts of U0 is OrderSortedSet of S1 by OSALG_1:17;
    then reconsider B as OSSubset of U0 by Def2;
    reconsider G = GenOSAlg(B) as Element of OSSub(U0) by Def14;
    reconsider G1 = G as Element of L;
    take G1;
    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(B)"\/"_os a2 by Def15
      .= G1 by Th38;
    hence thesis;
  end;
end;
