
theorem
  for V being RealUnitarySpace, W1,W2,W3 being strict Subspace of V
  holds W1 is Subspace of W2 implies W1 /\ W3 is Subspace of W2 /\ W3
proof
  let V be RealUnitarySpace, W1,W2,W3 be strict Subspace of V;
  set S = LattStr (# Subspaces(V), SubJoin(V), SubMeet(V) #);
  reconsider A = W1, B = W2, C = W3, AC = W1 /\ W3, BC = W2 /\ W3 as Element
  of S by Def3;
  assume
A1: W1 is Subspace of W2;
  A "\/" B = SubJoin(V).(A,B) by LATTICES:def 1
    .= W1 + W2 by Def7
    .= B by A1,Th8;
  then A [= B by LATTICES:def 3;
  then A "/\" C [= B "/\" C by LATTICES:9;
  then
A2: (A "/\" C) "\/" (B "/\" C) = (B "/\" C) by LATTICES:def 3;
A3: B "/\" C = SubMeet(V).(B,C) by LATTICES:def 2
    .= W2 /\ W3 by Def8;
  (A "/\" C) "\/" (B "/\" C) = SubJoin(V).(A "/\" C,B "/\" C) by LATTICES:def 1
    .= SubJoin(V).(SubMeet(V).(A,C),B "/\" C) by LATTICES:def 2
    .= SubJoin(V).(SubMeet(V).(A,C),SubMeet(V).(B,C)) by LATTICES:def 2
    .= SubJoin(V).(SubMeet(V).(A,C),BC) by Def8
    .= SubJoin(V).(AC,BC) by Def8
    .= (W1 /\ W3) + (W2 /\ W3) by Def7;
  hence thesis by A2,A3,Th8;
end;
