reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;
reserve n for Nat;

theorem
  for c,x be uSurreal st L_c << {x} << R_c & x <> c holds born c in born x
proof
  let c,x be uSurreal such that
  A1: L_c << {x} << R_c & x <> c and
  A2:not born c in born x;
  A3: born c = born_eq c by Th48;
  then born c c= born x c= born c by A1,A2, Th51,ORDINAL1:16;
  then A4:born c = born x by XBOOLE_0:def 10;
  not x == c by A1,Th50;
  then per cases;
  suppose x < c;
    then per cases by Th13;
    suppose ex xR be Surreal st xR in R_x & x < xR <= c;
      then consider xR be Surreal such that
      A5:xR in R_x & x < xR <= c;
      xR in L_x \/ R_x by A5,XBOOLE_0:def 3;
      then A6: born xR in born x by Th1;
      {c} << R_c & x <= xR by A5,Th11;
      then L_c << {xR} << R_c by A1,A5,Th17,Th18;
      then born c in born x by A3,Th51,A6,ORDINAL1:12;
      hence thesis by A4;
    end;
    suppose ex cL be Surreal st cL in L_c & x <= cL < c;
      then consider cL be Surreal such that
      A7:cL in L_c & x <= cL < c;
      L_c << {cL} & cL in {cL} by A1,TARSKI:def 1,A7,Th17;
      hence thesis by A7,Th3;
    end;
  end;
  suppose c < x;
    then per cases by Th13;
    suppose ex cR be Surreal st cR in R_c & c < cR <= x;
      then consider cR be Surreal such that
      A8:cR in R_c & c < cR <= x;
      cR in {cR} << R_c by A1,A8,Th18,TARSKI:def 1;
      hence thesis by A8,Th3;
    end;
    suppose ex xL be Surreal st xL in L_x & c <= xL < x;
      then consider xL be Surreal such that
      A9:xL in L_x & c <= xL < x;
      xL in L_x \/ R_x by A9,XBOOLE_0:def 3;
      then A10: born xL in born x by Th1;
      L_c << {c} & xL <= x by A9,Th11;
      then L_c << {xL} << R_c by A1,A9,Th17,Th18;
      then born c in born x by A3,Th51,A10,ORDINAL1:12;
      hence thesis by A4;
    end;
  end;
end;
