
theorem Th43:
  NATOrd is_antisymmetric_in NAT
proof
  let x, y be object such that x in NAT and y in NAT and
A1: [x,y] in NATOrd and
A2: [y,x] in NATOrd;
  consider x9, y9 being Element of NAT such that
A3: [x,y] = [x9,y9] and
A4: x9 <= y9 by A1;
A5: x = x9 by A3,XTUPLE_0:1;
A6: y = y9 by A3,XTUPLE_0:1;
  consider y2, x2 being Element of NAT such that
A7: [y,x] = [y2,x2] and
A8: y2 <= x2 by A2;
A9: y2 = y9 by A6,A7,XTUPLE_0:1;
  x2 = x9 by A5,A7,XTUPLE_0:1;
  hence thesis by A4,A5,A6,A8,A9,XXREAL_0:1;
end;
