reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
reserve M for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF;
reserve W,W1,W2,W3 for Subspace of M;
reserve u,u1,u2,v,v1,v2 for Element of M;
reserve X,Y for set, x,y,y1,y2 for object;

theorem
  W1 is Subspace of W2 implies W1 /\ W3 is Subspace of W2 /\ W3
proof
  set A1 = the carrier of W1;
  set A2 = the carrier of W2;
  set A3 = the carrier of W3;
  set A4 = the carrier of W1 /\ W3;
  assume W1 is Subspace of W2;
  then A1 c= A2 by VECTSP_4:def 2;
  then A1 /\ A3 c= A2 /\ A3 by XBOOLE_1:26;
  then A4 c= A2 /\ A3 by Def2;
  then A4 c= the carrier of W2 /\ W3 by Def2;
  hence thesis by VECTSP_4:27;
end;
