reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;
reserve d,d1,d2 for Dyadic;
reserve i,j for Integer,
        n,m,p for Nat;

theorem Th31:
  born x is finite & card (L_x) (+) card (R_x) c= 1 implies
    ex i be Integer st x == uInt.i
proof
  assume
A1: born x is finite & card (L_x) (+) card (R_x) c= 1;
A2: born x = born -x by SURREALR:12;
  per cases;
  suppose card (L_x) (+) card (R_x) = 1;
    then consider y be Surreal such that
A3:  x= [{},{y}] or x = [{y},{}] by SURREALO:47;
    per cases by A3;
    suppose x= [{},{y}];
      then -- R_x = --{y} ={-y} & -- L_x = --{} = {} by SURREALR:21,22;
      then
A4:    -x = [{-y},{}] by SURREALR:7;
      per cases;
      suppose -y < 0_No;
        then -x == 0_No by A4,Th16;
        then x = - -x == 0_No = uInt.0 by Def1,SURREALR:23,10;
        hence thesis;
      end;
      suppose 0_No<=-y;
        then consider n be Nat such that
A5:     -x == uInt.(n+1) & uInt.n <= -y < uInt.(n+1) &
        n in born x by A4,A1,A2,Th17;
        x =- -x == -uInt.(n+1) = uInt.-(n+1) by A5,SURREALR:10,Th12;
        hence thesis;
      end;
    end;
    suppose
A6:   x= [{y},{}];
      per cases;
      suppose y < 0_No;
        then x == 0_No = uInt.0 by Def1,A6,Th16;
        hence thesis;
      end;
      suppose 0_No<=y;
        then ex n be Nat st x == uInt.(n+1) & uInt.n <= y < uInt.(n+1) &
        n in born x by A6,A1,Th17;
        hence thesis;
      end;
    end;
  end;
  suppose card (L_x) (+) card (R_x) <> 1;
    then card (L_x) (+) card (R_x) in 1=succ 0
    by A1,XBOOLE_0:def 8,ORDINAL1:11;
    then card (L_x) (+) card (R_x) = 0;
    then x = 0_No by SURREALO:46;
    then x = uInt.0 by Def1;
    hence thesis;
  end;
end;
