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 Th51:
  for c be Surreal st born c = born_eq c &
    L_c << {x} << R_c holds born c c= born x
proof
  let c be Surreal such that
  A1: born c = born_eq c & L_c << {x} << R_c and
  A2:not born c c= born x;
  defpred P[Ordinal] means  ex y st L_c << {y} << R_c & born y = $1;
  P[born x] by A1;
  then A3:ex A st P[A];
  consider A such that
  A4: P[A] & for B st P[B] holds A c= B from ORDINAL1:sch 1(A3);
  consider y such that
  A5: L_c << {y} << R_c & born y = A by A4;
  A c= born x in born c by A1,A2,ORDINAL1:16,A4;
  then A6: born y in born c by A5,ORDINAL1:12;
  for z st L_c << {z} << R_c holds born y c= born z by A4,A5;
  then born_eq c = born_eq y by A5,Th16,Th33;
  hence thesis by A1,ORDINAL1:5,A6,Def5;
end;
