reserve x,y,y1,y2,z,e,s for set;
reserve alpha,beta,gamma for Ordinal;
reserve n,m,k for Nat;

theorem Th3:
  alpha c= beta implies ConwayDay(alpha) c= ConwayDay(beta)
proof
  assume
A1: alpha c= beta;
  let z be object;
  assume z in ConwayDay(alpha);
  then consider w being strict left-right such that
A2: w = z & (for e st e in (the LeftOptions of w) \/ (the RightOptions of w)
      ex gamma st gamma in alpha & e in ConwayDay(gamma)) by Th1;
  now
    let e;
    assume e in (the LeftOptions of w) \/ (the RightOptions of w);
    then ex gamma st gamma in alpha & e in ConwayDay(gamma) by A2;
    hence ex gamma st gamma in beta & e in ConwayDay(gamma) by A1;
  end;
  hence z in ConwayDay(beta) by Th1,A2;
end;
