reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;

theorem LMTFRat2:
  for x being Integer, v being Element of F_Rat, v1 being Rational
  st v = v1 holds (Int-mult-left(F_Rat)).(x,v) = x*v1
  proof
    let x be Integer, v be Element of F_Rat, v1 be Rational;
    assume A1: v = v1;
    reconsider xx = x as Element of INT.Ring by INT_1:def 2;
    per cases;
    suppose C1: x >= 0;
      then reconsider x0 = x as Element of NAT by INT_1:3;
      thus (Int-mult-left(F_Rat)).(x,v) = (Nat-mult-left F_Rat).(xx,v)
      by C1,ZMODUL01:def 20
      .= x0*v1 by LMTFRat1,A1
      .= x*v1;
    end;
    suppose C2: x < 0;
      then reconsider x0 = -x as Element of NAT by INT_1:3;
      thus (Int-mult-left(F_Rat)).(x,v)
      = (Nat-mult-left F_Rat) . (-xx,-v) by C2,ZMODUL01:def 20
      .= x0*(-v1) by A1,LMTFRat1
      .= x*v1;
    end;
  end;
