 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem ThRankS1:
  for V being Z_Module, W being finite-rank free Subspace of V,
  a being Element of INT.Ring st a <> 0.INT.Ring holds
  rank(a (*) W) = rank(W)
  proof
    let V be Z_Module, W be finite-rank free Subspace of V,
    a be Element of INT.Ring such that
    A1: a <> 0.INT.Ring;
    defpred P[Element of W,object] means $2 = a * $1;
    P0: for x being Element of W ex y being Element of a (*) W st P[x,y]
    proof
      let x be Element of W;
      a * x in the carrier of (a (*) W);
      hence thesis;
    end;
    consider F be Function of W,a (*) W such that
    A2: for x be Element of W holds P[x,F.x] from FUNCT_2:sch 3(P0);
    X1X: for x1, x2 being object st x1 in the carrier of W
    & x2 in the carrier of W & F.x1 = F.x2
    holds x1 = x2
    proof
      let x1, x2 be object such that
      D1: x1 in the carrier of W & x2 in the carrier of W  & F.x1 = F.x2;
      reconsider xx1 = x1, xx2 = x2 as Element of W by D1;
      F.x1 = a*xx1 & F.x2= a*xx2 by A2;
      hence thesis by A1,ZMODUL01:10,D1;
    end;
    for y being object st y in the carrier of a (*) W holds y in rng F
    proof
      let y be object such that
      D1: y in the carrier of a (*) W;
      consider v be Element of W such that
      D2: y=a*v by D1;
      F.v = a*v by A2;
      hence y in rng F by D2,FUNCT_2:4;
    end;
    then X2X: the carrier of a (*) W c= rng F;
    B0: F is additive
    proof
      let x, y be Element of W;
      B03: F.x = a*x by A2;
      B04: F.y = a*y by A2;
      thus F.(x+y) = a*(x+y) by A2
      .= a*x + a*y by VECTSP_1:def 14
      .= F.x + F.y by ZMODUL01:28,B03,B04;
    end;
    for r be Element of INT.Ring, x be Element of W holds F.(r*x) = r*F.x
    proof
      let r be Element of INT.Ring, x be Element of W;
      thus F.(r*x) = a*(r*x) by A2
      .= (a*r)*x by VECTSP_1:def 16
      .= r*(a*x) by VECTSP_1:def 16
      .= r*F.x by ZMODUL01:29,A2;
    end;
    then reconsider F as linear-transformation of W, a (*) W
      by B0,MOD_2:def 2;
    reconsider aW = a (*) W as finite-rank free Z_Module;
    reconsider W0 = W as finite-rank free Z_Module;
    reconsider F as linear-transformation of W0, aW;
    F is one-to-one onto by X1X,FUNCT_2:19,X2X,FUNCT_2:def 3,XBOOLE_0:def 10;
    hence thesis by HM15;
  end;
