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

theorem Th2:
  x in Day n implies uInt.-n <= x <= uInt.n
proof
  defpred P[Nat] means for x st x in Day $1 holds
  uInt.-$1 <= x <= uInt.$1;
A1: P[0]
  proof
A2: uInt.0=0_No by Def1;
    let x;
    assume x in Day 0;
    then born x c= 0 by SURREAL0:def 18;
    then born x = 0;
    hence thesis by A2,SURREAL0:37;
  end;
A3: for n holds P[n] implies P[n+1]
  proof
    let n;
A4: succ Segm n = Segm (n+1) by NAT_1:38;
    assume
A5: P[n];
    let x such that
A6: x in Day (n+1);
    uInt.(n+1) = [{uInt.n},{}] by Def1;
    then
A7: R_(uInt.(n+1)) = {};
A8: L_x << {uInt.(n+1)}
    proof
      let l,r be Surreal such that
A9:   l in L_x & r in {uInt.(n+1)};
      l in L_x \/ R_x by A9,XBOOLE_0:def 3;
      then born l in born x c= n+1 by A6,SURREALO:1,SURREAL0:def 18;
      then l in Day born l c= Day n by A4,SURREAL0:35,def 18,ORDINAL1:22;
      then l <= uInt.n < uInt.(n+1) by A5,Lm1;
      then l < uInt.(n+1) by SURREALO:4;
      hence thesis by A9,TARSKI:def 1;
    end;
A10: {x} << {};
    uInt.-(n+1) = [{},{uInt.-n}] by Def1;
    then
A11: L_(uInt.-(n+1)) ={};
A12: {} << {x};
    {uInt.-(n+1)} << R_x
    proof
      let r,l be Surreal such that
A13:  r in {uInt.-(n+1)} & l in R_x;
      l in L_x \/ R_x by A13,XBOOLE_0:def 3;
      then born l in born x c= n+1 by A6,SURREALO:1,SURREAL0:def 18;
      then l in Day born l c= Day n by A4,ORDINAL1:22,SURREAL0:35,def 18;
      then uInt.-(n+1) < uInt.-n <= l by A5,Lm2;
      then uInt.-(n+1) < l by SURREALO:4;
      hence thesis by A13,TARSKI:def 1;
    end;
    hence thesis by A7,A8,SURREAL0:43,A10,A11,A12;
  end;
  for n holds P[n] from NAT_1:sch 2(A1,A3);
  hence thesis;
end;
