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

theorem
  y in born_eq_set x implies born y = born_eq y = born_eq x
proof
  assume A1: y in born_eq_set x;
  then y in Day born_eq x by Def6;
  then A2:born y c= born_eq x by SURREAL0:def 18;
  x==y by A1,Def6;
  then A3:born_eq x = born_eq y by Th33;
  born_eq y c= born y by Def5;
  hence thesis by A2,A3,XBOOLE_0:def 10;
end;
