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

theorem Th7:
  for x be Element of Games O holds
    x in Day(R,O) iff L_x <<R, R_x &
      for o st o in L_x \/ R_x
        ex A st A in O & o in Day(R,A)
proof
  let A be Element of Games O;
  consider L be Sequence such that
  A1:Day(R,O) = L.O & dom L = succ O and
  A2: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} by Def6;
  A3: O in succ O by ORDINAL1:6;
  A4: Day(R,O) ={x where x is Element of Games O:
  L_x c= union rng (L|O) & R_x c= union rng (L|O) & L_x <<R, R_x}
    by A1,A2,ORDINAL1:6;
  A5:dom (L|O) = O by RELAT_1:62,A1,A3,ORDINAL1:def 2;
  thus A in Day(R,O) implies L_A <<R, R_A &
  for x be object st x in L_A \/ R_A
    ex B be Ordinal st B in O & x in Day(R,B)
  proof
    assume A in Day(R,O);
    then consider x be Element of Games O such that
    A6:A=x& L_x c= union rng (L|O) & R_x c= union rng (L|O) &
    L_x <<R, R_x by A4;
    thus L_A <<R, R_A by A6;
    let y be object such that A7:y in L_A \/ R_A;
    L_x\/R_x c= union rng (L|O) by A6,XBOOLE_1:8;
    then consider Y be set such that
    A8: y in Y & Y in rng (L|O) by A6,A7,TARSKI:def 4;
    consider B be object such that
    A9: B in dom (L|O) & (L|O).B=Y by A8,FUNCT_1:def 3;
    reconsider B as Ordinal by A9;
    take B;
    A10:B in succ O by A9,ORDINAL1:8;
    (L|O).B = L.B by A9,FUNCT_1:49;
    hence thesis by A9,A1,A8,A10,A2,Th6;
  end;
  assume that A11: L_A <<R, R_A and
  A12: for x be object st x in L_A \/ R_A
  ex B be Ordinal st B in O & x in Day(R,B);
  A13: L_A \/ R_A c= union rng (L|O)
  proof
    let x be object such that A14:x in L_A \/ R_A;
    consider B be Ordinal such that
    A15: B in O & x in Day(R,B) by A14,A12;
    B in succ O by A15,ORDINAL1:8;
    then Day(R,B) = L.B = (L|O).B in rng (L|O)
      by A5,A15,A1,A2,Th6,FUNCT_1:49,def 3;
    hence thesis by A15,TARSKI:def 4;
  end;
  L_A c= L_A \/ R_A & R_A c= L_A \/ R_A by XBOOLE_1:7;
  then L_A c= union rng (L|O) & R_A c= union rng (L|O) by A13,XBOOLE_1:1;
  hence thesis by A11,A4;
end;
