reserve A,A1,A2,B,B1,B2,C,O for Ordinal,
      R,S for Relation,
      a,b,c,o,l,r for object;

theorem Th9:
   A c= B implies Day(R,A) c= Day(R,B)
proof
  assume A1:A c= B;
  let x1,x2 be object;
  set x=[x1,x2];
  assume A2:x in Day(R,A);
  then A3:x in Games A;
  A4:L_x <<R, R_x &
      for y be object st y in L_x \/ R_x
        ex O be Ordinal st O in A & y in Day(R,O) by Th7,A2;
  A5:Games A c= Games B by A1,Th1;
  for y be object st y in L_x \/ R_x
    ex O be Ordinal st O in B & y in Day(R,O)
  proof
    let y be object;
    assume y in L_x \/ R_x;
    then ex O be Ordinal st O in A & y in Day(R,O) by Th7,A2;
    hence thesis by A1;
  end;
  hence thesis by A4,A5,A3, Th7;
end;
