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

theorem Th12:
  o in Games O & not o in Day(R,O) implies not o in Day(R,A)
proof
  defpred P[Ordinal] means for x be object, O be Ordinal st
    x in (Games O) \  Day(R,O) holds not x in Day(R,$1);
  A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
  proof
    let D be Ordinal such that A2: for C be Ordinal st C in D holds P[C];
    let x be object;
    let O be Ordinal such that
    A3: x in (Games O) \  Day(R,O) and
    A4: x in Day(R,D);
    reconsider xD=x as Element of Games D by A4;
    reconsider xO=x as Element of Games O by A3;
    A5: L_xD <<R, R_xD & for x be object st x in L_xD \/ R_xD
      ex O be Ordinal st O in D & x in Day(R,O) by A4,Th7;
    not xO in Day(R,O) by A3,XBOOLE_0:def 5;
    then consider y be object such that
    A6:  y in L_xO \/ R_xO & for H be Ordinal st H in O holds
    not y in Day(R,H) by A5,Th7;
    consider H be Ordinal such that
    A7: H in O & y in Games H by A6,Th4;
    not y in Day(R,H) by A6,A7;
    then A8: y in (Games H)\Day(R,H) by A7,XBOOLE_0:def 5;
    ex W be Ordinal st
    W in D & y in Day(R,W) by A6,A4,Th7;
    hence thesis by A8,A2;
  end;
  A9:for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
  assume o in Games O & not o in Day(R,O);
  then o in (Games O) \ Day(R,O) by XBOOLE_0:def 5;
  hence thesis by A9;
end;
