 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 LMINTRNG2:
  for x be Element of INT.Ring, v be Element of INT.Ring,
      v1 be Integer
  st v = v1 holds (Int-mult-left INT.Ring).(x,v) = x*v1
  proof
    let x be Element of INT.Ring, v be Element of INT.Ring,
        v1 be Integer;
    assume A1: v = v1;
    reconsider xx = x as Element of INT;
    per cases;
    suppose C1: x >= 0;
      then reconsider x0 = x as Element of NAT by INT_1:3;
      thus (Int-mult-left INT.Ring).(x,v) = (Nat-mult-left INT.Ring).(xx,v)
        by C1,ZMODUL01:def 20
      .= (Nat-mult-left INT.Ring).(x0,v)
      .= x0*v1 by LMINTRNG1,A1
      .= x*v1;
    end;
    suppose C2: x < 0;
      then reconsider x0 = -x as Element of NAT by INT_1:3;
      thus (Int-mult-left(INT.Ring)).(x,v)
      = (Nat-mult-left INT.Ring) . (-x,-v) by C2,ZMODUL01:def 20
      .= x0*(-v1) by LMINTRNG1,A1
      .= x*v1;
    end;
  end;
