reserve N,M,K for ExtNat;

theorem
  K <> 0 & N = M * K implies M <= N
proof
  assume A1: K <> 0 & N = M * K;
  per cases by Th3;
  suppose N is Nat;
    then reconsider n = N as Nat;
    per cases;
    suppose M = 0;
      hence thesis;
    end;
    suppose A2: M <> 0;
      M <> +infty & K <> +infty
      proof
        assume M = +infty or K = +infty;
        then n = +infty by A1, A2, XXREAL_3:def 5;
        hence contradiction;
      end;
      then reconsider m = M, k = K as Nat by Th3;
      n = m * (k qua ExtNat) by A1;
      hence thesis by A1, NAT_1:24;
    end;
  end;
  suppose N = +infty;
    hence thesis by XXREAL_0:3;
  end;
end;
