reserve N,M,K for ExtNat;

theorem Th5:
  M in NAT & N <= M implies N in NAT
proof
  assume A1: M in NAT & N <= M;
  0 <= N & 0 in REAL by XREAL_0:def 1;
  then N in REAL by A1, NUMBERS:19, XXREAL_0:45;
  then N is Nat by Th3;
  hence thesis by ORDINAL1:def 12;
end;
