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 Th16:
  (for W1 being strict Subspace of M holds W1 is Subspace of W2
  implies W1 /\ W2 = W1) & for W1 st W1 /\ W2 = W1 holds W1 is Subspace of W2
proof
  thus for W1 being strict Subspace of M holds W1 is Subspace of W2 implies W1
  /\ W2 = W1
  proof
    let W1 be strict Subspace of M;
    assume W1 is Subspace of W2;
    then
A1: the carrier of W1 c= the carrier of W2 by VECTSP_4:def 2;
    the carrier of W1 /\ W2 = (the carrier of W1) /\ (the carrier of W2)
    by Def2;
    hence thesis by A1,VECTSP_4:29,XBOOLE_1:28;
  end;
  thus thesis by Th15;
end;
