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

theorem Th33:
   x==y implies born_eq x = born_eq y
proof
  assume A1:x==y;
  consider z be Surreal such that
  A2:born z= born_eq x & z ==x by Def5;
  z==y by A1,A2,Th4;
  then A3: born_eq y c= born_eq x by A2,Def5;
  consider t be Surreal such that
  A4:born t= born_eq y & t ==y by Def5;
  t==x by A1,A4,Th4;
  then born_eq x c= born_eq y by A4,Def5;
  hence thesis by A3,XBOOLE_0:def 10;
end;
