reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;

theorem Th5:
  born_eq uInt.n = n & born_eq uInt.-n = n
proof
A1: born uInt.n = n by Th4;
  for y be Surreal st y == uInt.n holds born uInt.n c= born y
  proof
    let y be Surreal such that
A2: y == uInt.n;
    assume not born uInt.n c= born y;
    then
A3: not n c= born y by Th4;
    then reconsider O=born y as Nat;
    y in Day O c= Day n by SURREAL0:35,SURREAL0:def 18,A3;
    then y <= uInt.O by Th2;
    then
A4: uInt.n <= uInt.O by A2,SURREALO:4;
    O in Segm n by A3,ORDINAL1:16;
    hence thesis by A4,Th3,NAT_1:44;
  end;
  hence born_eq uInt.n = n by A1,SURREALO:def 5;
A5: born uInt.-n = n by Th4;
  for y be Surreal st y == uInt.-n holds born uInt.-n c= born y
  proof
    let y be Surreal such that
A6: y == uInt.-n;
    assume not born uInt.-n c= born y;
    then
A7: not n c= born y by Th4;
    then reconsider O=born y as Nat;
    y in Day O c= Day n by SURREAL0:35,SURREAL0:def 18,A7;
    then
A8: uInt.-O <= y by Th2;
    O in Segm n by A7,ORDINAL1:16;
    then -n < -O by NAT_1:44,XREAL_1:24;
    hence thesis by A8, A6,SURREALO:4,Th3;
  end;
  hence thesis by A5,SURREALO:def 5;
end;
