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 LMTFRat1:
  for v being Element of F_Rat,v1 be Rational
  st v = v1 holds
  for n being Nat holds (Nat-mult-left(F_Rat)).(n,v) = n*v1
  proof
    let v be Element of F_Rat,v1 be Rational;
    assume A1: v = v1;
    defpred P[Nat] means
    (Nat-mult-left(F_Rat)).($1,v) =$1*v1;
    (Nat-mult-left(F_Rat)).(0,v) = 0.F_Rat by BINOM:def 3
    .= 0 * v1;
    then
    X1: P[0];
    X2: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume X22: P[n];
      (Nat-mult-left(F_Rat)).(n+1,v) = v + (Nat-mult-left(F_Rat)).(n,v)
      by BINOM:def 3
      .= v1 + n*v1 by A1,X22
      .= (n+1)*v1;
      hence thesis;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(X1,X2);
    hence thesis;
  end;
