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

theorem ThEQRZMV3E:
  for V be free Z_Module,
  I being Subset of V,
  IQ being Subset of Z_MQ_VectSp(V)
  st IQ =(MorphsZQ(V)).:I
  holds card(I) = card(IQ)
  proof
    let V be free Z_Module,
    I be Subset of V,
    IQ be Subset of Z_MQ_VectSp(V);
    assume AS1: IQ =(MorphsZQ(V)).:I;
    P1: MorphsZQ(V) is one-to-one by defMorph;
    the carrier of V = dom (MorphsZQ(V)) by FUNCT_2:def 1;
    hence card(I) = card(IQ) by AS1,P1,CARD_1:5,CARD_1:33;
  end;
