 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 LMINTRNG1:
  for v be Element of INT.Ring,v1 be Integer st v = v1 holds
  for n be Nat holds (Nat-mult-left(INT.Ring)).(n,v) = n*v1
  proof
    let v be Element of INT.Ring,v1 be Integer;
    assume A1: v = v1;
    defpred P[Nat] means
    (Nat-mult-left(INT.Ring)).($1,v) = $1*v1;
    (Nat-mult-left(INT.Ring)).(0,v) = 0.(INT.Ring) by BINOM:def 3
    .= (0.INT.Ring) * v1; then
    X1: P[0];
    X2: for n be Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume X22: P[n];
      (Nat-mult-left(INT.Ring)).(n+1,v) = v + (Nat-mult-left(INT.Ring)).(n,v)
      by BINOM:def 3
      .= (n+1)*v1 by A1,X22;
      hence thesis;
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(X1,X2);
    hence thesis;
  end;
