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

theorem Th6:
  for L be Sequence st
    dom L = succ O &
    for A st A in succ O holds
      L.A = {x where x is Element of Games A:
             L_x c= union rng (L|A) & R_x c= union rng (L|A) & L_x <<R, R_x}
   holds for A st A in succ O holds L.A = Day(R,A)
proof
  let L be Sequence such that
  A1:dom L = succ O and
  A2:for A be Ordinal st A in succ O holds
  L.A = {x where x is Element of Games A:
    L_x c= union rng (L|A) & R_x c= union rng (L|A) & L_x <<R, R_x};
  let D be Ordinal such that A3: D in succ O;
  consider LO be Sequence such that
  A4:Day(R,D) = LO.D & dom LO = succ D and
  A5:for A be Ordinal st A in succ D holds
  LO.A = {x where x is Element of Games A:
  L_x c= union rng (LO|A) & R_x c= union rng (LO|A) & L_x <<R, R_x} by Def6;
  defpred P[Ordinal] means $1 c= D 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= D;
    then A9:A in succ D by ORDINAL1:22;
    A10: A in succ O by A8,A3,ORDINAL1:12;
    A11: L.A = {x where x is Element of Games A:
    L_x c= union rng (L|A) & R_x c= union rng (L|A) &
    L_x <<R, R_x} 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 A8,ORDINAL1:22,A5,A11;
  end;
  for A holds P[A] from ORDINAL1:sch 2(A6);
  hence thesis by A4;
end;
