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;

theorem Th17:
  for W1 being strict Subspace of V holds W1 is Subspace of W2 iff
  W1 /\ W2 = W1
proof
  let W1 be strict Subspace of V;
  thus W1 is Subspace of W2 implies W1 /\ W2 = W1
  proof
    assume W1 is Subspace of W2;
    then
A1: the carrier of W1 c= the carrier of W2 by RLSUB_1:def 2;
    the carrier of W1 /\ W2 = (the carrier of W1) /\ (the carrier of W2 )
    by Def2;
    hence thesis by A1,RLSUB_1:30,XBOOLE_1:28;
  end;
  thus thesis by Th16;
end;
