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

theorem Th4:
  n = born uInt.n & n = born uInt.-n
proof
A1: uInt.n in Day n by Th1;
  for O st uInt.n in Day O holds n c= O
  proof
    let O such that
A2: uInt.n in Day O and
A3: not n c= O;
A4: O in Segm n by A3,ORDINAL1:16;
    reconsider O as Nat by A3;
    uInt.n <= uInt.O < uInt.n by A2,Th2,A4,NAT_1:44,Th3;
    hence thesis;
  end;
  hence n = born (uInt.n) by A1,SURREAL0:def 18;
A5: uInt.-n in Day n by Th1;
  for O st uInt.-n in Day O holds n c= O
  proof
    let O such that
A6: uInt.-n in Day O and
A7: not n c= O;
A8: O in Segm n by A7,ORDINAL1:16;
    reconsider O as Nat by A7;
A9: -n < - O by XREAL_1:24,A8,NAT_1:44;
    uInt.-n <= uInt.-O < uInt.-n by A6,Th2,A9,Th3;
    hence thesis;
  end;
  hence thesis by A5,SURREAL0:def 18;
end;
