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 Th50:
  for U0 be non-empty OSAlgebra of S1, B be OSSubset of U0 st B =
  the Sorts of U0 holds Top (OSSubAlLattice(U0)) = GenOSAlg(B)
proof
  let U0 be non-empty OSAlgebra of S1, B be OSSubset of U0;
  reconsider G = GenOSAlg(B) as Element of OSSub(U0) by Def14;
  set L = OSSubAlLattice(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 OSSub(U0);
    reconsider a2 = a1 as strict OSSubAlgebra of U0 by Def14;
    thus G1"\/" a = GenOSAlg(B)"\/"_os a2 by Def15
      .= G1 by A1,Th38;
    hence a "\/" G1 = G1;
  end;
  hence thesis by LATTICES:def 17;
end;
