
theorem Th56:
  for V being RealUnitarySpace holds LattStr (# Subspaces(V),
    SubJoin(V), SubMeet(V) #) is upper-bounded
proof
  let V be RealUnitarySpace;
  set S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #);
  ex C being Element of S st for A being Element of S holds C "\/" A = C &
  A "\/" C = C
  proof
    reconsider C = (Omega).V as Element of S by Def3;
    take C;
    let A be Element of S;
    reconsider W = A as Subspace of V by Def3;
    thus C "\/" A = SubJoin(V).(C,A) by LATTICES:def 1
      .= (Omega).V + W by Def7
      .= the UNITSTR of V by Th11
      .= C by RUSUB_1:def 3;
    hence thesis;
  end;
  hence thesis by LATTICES:def 14;
end;
