
theorem Th1:
  NAT c< ExtNAT & ExtNAT c< ExtREAL
proof
  A2: +infty in ExtNAT by ZFMISC_1:136;
  not +infty in NAT;
  hence NAT c< ExtNAT by A2, XBOOLE_1:7, XBOOLE_0:def 8;
  -infty in {+infty,-infty} by TARSKI:def 2;
  then A2: -infty in ExtREAL by XXREAL_0:def 4, XBOOLE_0:def 3;
  not -infty in ExtNAT
  proof
    assume -infty in ExtNAT;
    then per cases by ZFMISC_1:136;
    suppose -infty in NAT;
      hence contradiction;
    end;
    suppose -infty in {+infty};
      hence contradiction by TARSKI:def 1;
    end;
  end;
  hence thesis by A2, XBOOLE_0:def 8;
end;
