
theorem Th44:
  NATOrd is_strongly_connected_in NAT
proof
  now
    let x, y be object such that
A1: x in NAT and
A2: y in NAT;
    thus [x,y] in NATOrd or [y,x] in NATOrd
    proof
      assume that
A3:   not [x,y] in NATOrd and
A4:   not [y,x] in NATOrd;
      reconsider x,y as Element of NAT by A1,A2;
      per cases;
      suppose x <= y;
        hence contradiction by A3;
      end;
      suppose y <= x;
        hence contradiction by A4;
      end;
    end;
  end;
  hence thesis;
end;
