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

theorem
  x in born_eq_set y implies -x in born_eq_set -y
proof
  assume A1:x in born_eq_set y;
  then x == y by SURREALO:def 6;
  then A2: -x == -y by Th10;
  x in Day born_eq y by A1,SURREALO:def 6;
  then born -x = born x c= born_eq y =born_eq -y
    by SURREAL0:def 18,Th13,Th12;
  then -x in Day (born -x) c= Day (born_eq -y) by SURREAL0:def 18,35;
  hence thesis by A2,SURREALO:def 6;
end;
