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

theorem Th17:
  x = [{y},{}] & born x is finite & 0_No <= y implies
    ex n be Nat st x == uInt.(n+1) & uInt.n <= y < uInt.(n+1) & n in born x
proof
  assume
A1: x=[{y},{}] & born x is finite & 0_No <= y;
  reconsider a = born x as Nat by A1;
  defpred O[Nat] means L_x << {uInt.$1};
A2: x in Day a by SURREAL0:def 18;
  L_x << {x} by SURREALO:11;
  then O[a] by A2,Th2,SURREALO:17;
  then
A3: ex k being Nat st O[k];
  consider k be Nat such that
A4: O[k] & for n being Nat st O[n] holds k <= n from NAT_1:sch 5(A3);
  k<>0
  proof
    assume k=0;
    then {y} << {0_No} by Def1,A1,A4;
    hence thesis by A1,SURREALO:21;
  end;
  then reconsider k1=k-1 as Nat by NAT_1:20;
  take k1;
A5: L_x << {uInt.k} << R_x by A4,A1;
  for z st L_x << {z} << R_x holds born uInt.k c= born z
  proof
    let z such that
A6: L_x << {z} << R_x and
A7: not born uInt.k c= born z;
A8: born z in born uInt.k =k=Segm k by Th4,A7,ORDINAL1:16;
    then reconsider bz = born z as Nat;
    bz < k = k1+1 by A8,NAT_1:44;
    then bz <= k1 by NAT_1:13;
    then Segm bz c= Segm k1 by NAT_1:39;
    then z in Day bz c= Day k1 by SURREAL0:35,SURREAL0:def 18;
    then L_x << {uInt.k1} by Th2,A6,SURREALO:17;
    then k1+1 <= k1 by A4;
    hence thesis by NAT_1:13;
  end;
  hence x == uInt.(k1+1) by A5,SURREALO:16;
A9: uInt.k1 <= y
  proof
    assume y < uInt.k1;
    then L_x << {uInt.k1} by SURREALO:21,A1;
    then k1+1 <= k1 by A4;
    hence thesis by NAT_1:13;
  end;
  hence uInt.k1 <= y < uInt.(k1+1) by SURREALO:21,A4,A1;
A10: k1 c= born y
  proof
    assume
A11:not k1 c= born y;
    then k1<>0;
    then reconsider k2=k1-1 as Nat by NAT_1:20;
    k1 = k2+1;
    then Segm k1 = succ Segm k2 by NAT_1:38;
    then born y c= k2 by A11,ORDINAL1:16,22;
    then y in Day born y c= Day k2 by SURREAL0:35,SURREAL0:def 18;
    then y <= uInt.k2 < uInt.(k2+1) by Th2,Lm1;
    hence thesis by A9,SURREALO:4;
  end;
  y in L_x \/ R_x by A1,TARSKI:def 1;
  hence thesis by SURREALO:1,A10,ORDINAL1:12;
end;
