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);
reserve l for Lattice;
reserve a,b for Element of l;

theorem
  for V being RealLinearSpace, 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 RealLinearSpace, 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 = W1 + W2 by Def7
    .= B by A1,Th8;
  then A [= B;
  then A "/\" C [= B "/\" C by LATTICES:9;
  then
A2: (A "/\" C) "\/" (B "/\" C) = (B "/\" C);
A3: B "/\" C = W2 /\ W3 by Def8;
  (A "/\" C) "\/" (B "/\" C) = 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;
