reserve N,M,K for ExtNat;

theorem
  K * (N+M) = K*N + K*M
proof
  per cases by Th3;
  suppose K is Nat;
    hence thesis by XXREAL_3:95;
  end;
  suppose A1: K = +infty;
    per cases;
    suppose A2: N is positive & M is positive;
      hence K * (N+M) = +infty by A1, XXREAL_3:def 5
        .= K*N + +infty by XXREAL_3:def 2
        .= K*N + K*M by A1, A2, XXREAL_3:def 5;
    end;
    suppose M is non positive;
      then A3: M = 0;
      hence K * (N+M) = K*N by XXREAL_3:4
        .= K*N + 0 by XXREAL_3:4
        .= K*N + K*M by A3;
    end;
    suppose N is non positive;
      then A3: N = 0;
      hence K * (N+M) = K*M by XXREAL_3:4
        .= 0 + K*M by XXREAL_3:4
        .= K*N + K*M by A3;
    end;
  end;
end;
