 reserve x, y, y1, y2 for set;
 reserve V for Z_Module;
 reserve u, v, w for Vector of V;
 reserve F, G, H, I for FinSequence of V;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve KL1, KL2 for Linear_Combination of V;
 reserve X for Subset of V;

theorem Th26:
  for p being prime Element of INT.Ring, V being free Z_Module,
      I being Basis of V holds
  card ( {ZMtoMQV(V,p,u) where u is Vector of V : u in I} )
  = card(I)
  proof
    let p be prime Element of INT.Ring, V be free Z_Module, I be Basis of V;
    set X = {ZMtoMQV(V,p,u) where u is Vector of V : u in I};
    set ZQ = Z_MQ_VectSp(V,p);
    per cases;
    suppose A1: I is empty;
      X = {}
      proof
        assume X <> {};
        then consider v0 be object such that
        A2: v0 in X by XBOOLE_0:def 1;
        consider u be Vector of V such that
        A3: v0=ZMtoMQV(V,p,u) & u in I by A2;
        thus contradiction by A3,A1;
      end;
      hence thesis by A1;
    end;
    suppose A4: I is non empty;
      now let x be object;
        assume x in X;
        then consider v be Vector of V such that
        A5: x = ZMtoMQV(V,p,v) & v in I;
        thus x in the carrier of ZQ by A5;
      end;
      then reconsider X as Subset of ZQ by TARSKI:def 3;
      consider v0 be object such that A6: v0 in I by A4,XBOOLE_0:def 1;
      reconsider v0 as Vector of V by A6;
      ZMtoMQV(V,p,v0) in X by A6;
      then reconsider X as non empty Subset of ZQ;
      consider F be Function of X, the carrier of V such that
      A7: (for u be Vector of V st u in I holds F.(ZMtoMQV(V,p,u)) = u)
      & F is one-to-one & dom F = X & rng F = I by Th25;
      thus thesis by A7,CARD_1:5,WELLORD2:def 4;
    end;
  end;
