reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem XXTh3:
  for V be free Z_Module,D,A be Subset of V
  st D is Basis of V & D is finite & card (D) c< card(A)
  holds A is linearly-dependent
  proof
    let V be free Z_Module,D,A be Subset of V;
    assume AS: D is Basis of V & D is finite & card (D) c< card(A);
    reconsider D0 = D as finite Subset of V by AS;
    A \ D0 <> {} by AS,CARD_1:68,ORDINAL1:11;
    then consider x be object such that
    P3: x in A \ D0 by XBOOLE_0:def 1;
    x in A & not x in D0 by P3,XBOOLE_0:def 5;
    then P5: card(D0 \/{x}) = card(D0) + 1 by CARD_2:41;
    succ card(D0) = card(D0)+^1 by ORDINAL2:31
    .= card(D0)+1 by CARD_2:36;
    then P6: card(D0)+1 c= card(A) by AS,ORDINAL1:11,ORDINAL1:21;
    consider f be Function such that
P7: f is one-to-one & dom f = (D0 \/{x} ) & rng f c= A by CARD_1:10,P5,P6;
    set B = rng f;
    P8: card (B) = card(D0)+1 by P5,P7,CARD_1:5,WELLORD2:def 4;
    then reconsider B as finite set;
    reconsider B as finite Subset of V by P7,XBOOLE_1:1;
    B is linearly-dependent by XXTh1,P8,AS;
    hence A is linearly-dependent by XXTh2,P7;
  end;
