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

theorem Th10:
  for i be Integer, x holds
     [{uInt.(i-1)},{uInt.(i+1)}] is Surreal &
     (x = [{uInt.(i-1)},{uInt.(i+1)}] implies x == uInt.i)
proof
  let i be Integer,x;
  set S=uInt.i;
  i+-1 < i+1 by XREAL_1:6;
  hence [{uInt.(i-1)},{uInt.(i+1)}] is Surreal by Th9,SURREALO:22;
  assume
A1:x = [{uInt.(i-1)},{uInt.(i+1)}];
  uInt.(i-1) < S by XREAL_1:44,Th9;
  then
A2: L_x << {S} by A1,SURREALO:21;
A3: L_S << {x} << R_S
  proof
    per cases;
    suppose i =0;
      then S = 0_No by Def1;
      hence thesis;
    end;
    suppose
A4:   i >0;
      then reconsider I=i as Element of NAT by INT_1:3;
      reconsider i1=I-1 as Nat by NAT_1:20,A4;
      i = i1+1;
      then
A5:   S = [{uInt.i1},{}] by Def1;
      uInt.i1 <= uInt.(i-1);
      then not L_x << {uInt.i1} by A1,SURREALO:21;
      then uInt.i1 < x by SURREAL0:43;
      hence thesis by SURREALO:21,A5;
    end;
    suppose
A6:   0 >i;
      then reconsider I=-i as Element of NAT by INT_1:3;
      reconsider i1=I-1 as Nat by NAT_1:20,A6;
      i = -(i1+1);
      then
A7:   S = [{},{uInt.-i1}] by Def1;
      not (L_(uInt.-i1) << {x} & {uInt.-i1} << R_x) by SURREALO:3,21,A1;
      then not uInt.-i1 <= x by SURREAL0:43;
      hence thesis by SURREALO:21,A7;
    end;
  end;
  S < uInt.(i+1) by Th9,XREAL_1:29;
  then {S} << R_x by A1,SURREALO:21;
  hence thesis by A2,A3,SURREAL0:43;
end;
