
theorem
  NATOrd = RelIncl omega
  proof
    set f = NATOrd, g = RelIncl omega;
    for a,b being object holds [a,b] in f iff [a,b] in g
    proof
      let a,b be object;
      hereby assume [a,b] in f; then
        consider x,y being Element of NAT such that
A1:     [a,b] = [x,y] & x <= y;
        x = Segm x & y = Segm y; then
        x c= y by A1,NAT_1:39;
        hence [a,b] in g by A1,WELLORD2:def 1;
      end;
      assume
A2:   [a,b] in g; then
      reconsider aa = a, bb = b as Element of NAT by ZFMISC_1:87;
A4:   aa c= bb by A2,WELLORD2:def 1;
      aa = Segm aa & bb = Segm bb; then
      aa <= bb by A4,NAT_1:39;
      hence thesis;
    end;
    hence thesis by RELAT_1:def 2;
  end;
