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

theorem Th4:
  o in Games O iff
     o is pair & for a st a in L_o \/ R_o
       ex A st A in O & a in Games A
proof
  consider L be Sequence such that
  A1:Games O= L.O & dom L = succ O and
  A2:for A st A in succ O holds
    L.A = [:bool union rng (L|A), bool union rng (L|A):] by Def4;
  A3: O in succ O by ORDINAL1:6;
  A4: Games O = [:bool union rng (L|O), bool union rng (L|O):]
    by A1,A2,ORDINAL1:6;
  A5:dom (L|O) = O by RELAT_1:62,A1,A3,ORDINAL1:def 2;
  thus o in Games O implies o is pair &
    for x be object st x in L_o \/ R_o
    ex B be Ordinal st B in O & x in Games B
  proof
    assume o in Games O;
    then consider x,y be object such that
    A6:x in bool union rng (L|O) & y in bool union rng (L|O) & o=[x,y]
      by A4,ZFMISC_1:def 2;
    reconsider x,y as set by TARSKI:1;
    thus o is pair by A6;
    let z be object such that A7:z in L_o \/ R_o;
    x\/y c= union rng (L|O) by A6,XBOOLE_1:8;
    then consider Y be set such that
    A8: z in Y & Y in rng (L|O) by A6,A7,TARSKI:def 4;
    consider OO be object such that
    A9: OO in dom (L|O) & (L|O).OO=Y by A8,FUNCT_1:def 3;
    reconsider OO as Ordinal by A9;
    take OO;
    A10:OO in succ O by ORDINAL1:8,A9;
    (L|O).OO = L.OO by A9,FUNCT_1:49;
    hence thesis by A9,A1,A8,A10,A2,Th3;
  end;
  assume that A11: o is pair and
  A12: for x be object st x in L_o \/ R_o
    ex B be Ordinal st B in O & x in Games B;
  A13: L_o \/ R_o c= union rng (L|O)
  proof
    let x be object such that A14:x in L_o \/ R_o;
    consider B be Ordinal such that
    A15: B in O & x in Games B by A14,A12;
    B in succ O by A15,ORDINAL1:8;
    then Games B = L.B = (L|O).B in rng (L|O)
    by A5,A15,A1,A2,Th3,FUNCT_1:49,def 3;
    hence thesis by A15,TARSKI:def 4;
  end;
  L_o c= L_o \/ R_o & R_o c= L_o \/ R_o by XBOOLE_1:7;
  then L_o c= union rng (L|O) & R_o c= union rng (L|O)
    by A13,XBOOLE_1:1;
  hence thesis by A11,A4,ZFMISC_1:def 2;
end;
