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

theorem Th1:
  uInt.n in Day n & uInt.-n in Day n
proof
  defpred P[Nat] means uInt.$1 in Day $1 & uInt.-$1 in Day $1;
A1:uInt.0 =0_No by Def1;
  born 0_No = {} by SURREAL0:37;
  then
A2: P[0] by A1,SURREAL0:def 18;
A3: for n st P[n] holds P[n+1]
  proof
    let n such that
A4: P[n];
A5: {uInt.n}<<{};
A6: n in succ Segm n = Segm (n+1) by ORDINAL1:6,NAT_1:38;
A7: {} << {uInt.-n};
A8:o in {uInt.-n}\/{} implies
    ex O st O in n+1 & o in Day O
    proof
      assume o in {uInt.-n}\/{};
      then o = uInt.-n by TARSKI:def 1;
      hence thesis by A6,A4;
    end;
    o in {uInt.n}\/{} implies ex O st O in n+1 & o in Day O
    proof
      assume o in {uInt.n}\/{};
      then o = uInt.n by TARSKI:def 1;
      hence thesis by A6,A4;
    end;
    then [{uInt.n},{}] in Day (n+1) &
    [{},{uInt.-n}] in Day (n+1) by A8,A5,A7,SURREAL0:46;
    hence thesis by Def1;
  end;
  for n holds P[n] from NAT_1:sch 2(A2,A3);
  hence thesis;
end;
