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 Th8:
  for W2 being strict Subspace of V holds W1 is Subspace of W2 iff W1 + W2 = W2
proof
  let W2 be strict Subspace of V;
  thus W1 is Subspace of W2 implies W1 + W2 = W2
  proof
    assume W1 is Subspace of W2;
    then the carrier of W1 c= the carrier of W2 by RLSUB_1:def 2;
    hence thesis by Lm3;
  end;
  thus thesis by Th7;
end;
