
theorem Th55:
  for V being RealUnitarySpace holds LattStr (# Subspaces(V),
    SubJoin(V), SubMeet(V) #) is lower-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 = (0).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 = SubMeet(V).(C,A) by LATTICES:def 2
      .= (0).V /\ W by Def8
      .= C by Th18;
    hence thesis;
  end;
  hence thesis by LATTICES:def 13;
end;
