reserve N,M,K for ExtNat;

theorem
  K < K + N iff 1 <= N & K <> +infty
proof
  hereby
    assume A1: K < K + N;
    assume N < 1 or K = +infty;
    then N = 0 or K = +infty by Th4c;
    hence contradiction by A1, XXREAL_3:4, XXREAL_3:def 2;
  end;
  assume A2: 1 <= N & K <> +infty;
  then reconsider k = K as Nat by Th3;
  per cases by Th3;
  suppose N is Nat;
    then reconsider n = N as Nat;
    k < (k qua ExtNat) + n by A2, NAT_1:19;
    hence thesis;
  end;
  suppose N = +infty;
    then A3: K + N = +infty by XXREAL_3:def 2;
    k in REAL by XREAL_0:def 1;
    hence thesis by A3, XXREAL_0:9;
  end;
end;
