reserve V for RealLinearSpace;
reserve W,W1,W2,W3 for Subspace of V;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a,a1,a2 for Real;
reserve X,Y,x,y,y1,y2 for set;
reserve C for Coset of W;
reserve C1 for Coset of W1;
reserve C2 for Coset of W2;
reserve t1,t2 for Element of [:the carrier of V, the carrier of V:];
reserve A1,A2,B for Element of Subspaces(V);

theorem Th55:
  for V being RealLinearSpace holds LattStr (# Subspaces(V),
    SubJoin(V), SubMeet(V) #) is lower-bounded
proof
  let V be RealLinearSpace;
  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 = (0).V /\ W by Def8
      .= C by Th18;
    hence thesis;
  end;
  hence thesis;
end;
