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

theorem Th3:
  for L be Sequence,O st
      dom L = succ O &
      for A st A in succ O holds
          L.A = [:bool union rng (L|A), bool union rng (L|A):]
  holds for A st A in succ O holds L.A = Games A
proof
  let L be Sequence, Ord be Ordinal such that
  A1:dom L = succ Ord and
  A2:for A be Ordinal st A in succ Ord holds
    L.A = [:bool union rng (L|A), bool union rng (L|A):];
  let O be Ordinal such that A3: O in succ Ord;
  consider LO be Sequence such that
  A4:Games O = LO.O & dom LO = succ O and
  A5:for A be Ordinal st A in succ O holds
    LO.A = [:bool union rng (LO|A), bool union rng (LO|A):] by Def4;
  defpred P[Ordinal] means $1 c= O implies LO.$1=L.$1;
  A6: for A be Ordinal st for C be Ordinal st C in A holds P[C] holds P[A]
  proof
    let A be Ordinal such that A7:for C be Ordinal st C in A holds P[C];
    assume A8: A c= O;
    then A9:A in succ O by ORDINAL1:22;
    A10: A in succ Ord by A8,A3,ORDINAL1:12;
    A11: L.A = [:bool union rng (L|A), bool union rng (L|A):]
      by A2,A8,A3,ORDINAL1:12;
    A12:dom (L|A)=A & dom (LO|A)=A by A4,A1,A9,ORDINAL1:def 2,RELAT_1:62,A10;
    for x be object st x in A holds (LO|A).x = (L|A).x
    proof
      let x be object such that A13:x in A;
      reconsider x as Ordinal by A13;
      (LO|A).x = LO.x by A13,FUNCT_1:49
      .= L.x by A13,A7,A8,ORDINAL1:def 2;
      hence thesis by A13,FUNCT_1:49;
    end;
    then
    LO|A = L|A by A12,FUNCT_1:2;
    hence thesis by A5,A8,ORDINAL1:22,A11;
  end;
  for A be Ordinal holds P[A] from ORDINAL1:sch 2(A6);
  hence thesis by A4;
end;
