
theorem Th45:
  NATOrd is_transitive_in NAT
proof
  let x, y, z be object such that x in NAT and y in NAT
  and z in NAT and
A1: [x,y] in NATOrd and
A2: [y,z] in NATOrd;
  consider x1,y1 being Element of NAT such that
A3: [x,y] = [x1,y1] and
A4: x1 <= y1 by A1;
A5: x = x1 by A3,XTUPLE_0:1;
A6: y = y1 by A3,XTUPLE_0:1;
  consider y2, z2 being Element of NAT such that
A7: [y,z] = [y2,z2] and
A8: y2 <= z2 by A2;
A9: y = y2 by A7,XTUPLE_0:1;
A10: z = z2 by A7,XTUPLE_0:1;
  x1 <= z2 by A4,A6,A8,A9,XXREAL_0:2;
  hence thesis by A5,A10;
end;
