 reserve R for Ring;
 reserve x, y, y1 for set;
 reserve a, b for Element of R;
 reserve V for LeftMod of R;
 reserve v, w for Vector of V;

theorem Th1:
  for V being Z_Module
  for a being Element of INT.Ring,
      v being Vector of V holds
    a = 0.INT.Ring or v = 0.V implies a * v = 0.V
  proof
    let V be Z_Module;
    let a be Element of INT.Ring,
        v be Vector of V;
    set R = INT.Ring;
    set N1=1.INT.Ring, N0=0.INT.Ring;
    assume
    A1: a = 0.R or v = 0.V;
    now
      per cases by A1;
      suppose
        A2: a = 0.R;
        v + N0 * v = N1 * v + N0 * v
        .= (N1 + N0) * v by VECTSP_1:def 15
        .= v
        .= v + 0.V by RLVECT_1:4;
        hence thesis by A2,RLVECT_1:8;
      end;
      suppose
        A3: v = 0.V;
        a * 0.V + a * 0.V = a * (0.V + 0.V) by VECTSP_1:def 14
        .= a * 0.V by RLVECT_1:4
        .= a * 0.V + 0.V by RLVECT_1:4;
        hence thesis by A3,RLVECT_1:8;
      end;
    end;
    hence thesis;
  end;
