theorem Th10:
  g in ConwayDay(alpha) & x in the_Options_of g implies x in ConwayDay(alpha)
proof
  assume g in ConwayDay(alpha) & x in the_Options_of g;
  then consider beta such that
A1: beta in alpha & x in ConwayDay(beta) by Th9;
  beta c= alpha by A1,ORDINAL1:def 2;
  then ConwayDay(beta) c= ConwayDay(alpha) by Th3;
  hence x in ConwayDay(alpha) by A1;
end;
