
theorem Th15:
  for V being RealUnitarySpace, W1,W2,W3 being Subspace of V holds
  W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3
proof
  let V be RealUnitarySpace;
  let W1,W2,W3 be Subspace of V;
  set V1 = the carrier of W1;
  set V2 = the carrier of W2;
  set V3 = the carrier of W3;
  the carrier of W1 /\ (W2 /\ W3) = V1 /\ (the carrier of W2 /\ W3) by Def2
    .= V1 /\ (V2 /\ V3) by Def2
    .= (V1 /\ V2) /\ V3 by XBOOLE_1:16
    .= (the carrier of W1 /\ W2) /\ V3 by Def2;
  hence thesis by Def2;
end;
