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

theorem Th1:
   A c= B implies Games A c= Games B
proof
  assume A1: A c= B;
    consider La be Sequence such that
    A2:Games A = La.A & dom La = succ A &
      for O be Ordinal st O in succ A holds
      La.O = [:bool union rng (La|O), bool union rng (La|O):] by Def4;
    consider Lb be Sequence such that
    A3:Games B = Lb.B & dom Lb = succ B &
      for O be Ordinal st O in succ B holds
      Lb.O = [:bool union rng (Lb|O), bool union rng (Lb|O):] by Def4;
    defpred P[Ordinal] means $1 c= A implies La.$1=Lb.$1;
    A4: 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 A5: for C be Ordinal st C in D holds P[C];
      assume A6: D c= A;
      B c= succ B by XBOOLE_1:7;
      then A c= succ A & A c= succ B by XBOOLE_1:7,A1,XBOOLE_1:1;
      then A7:dom (La|D)=D = dom (Lb|D)
        by A2,A3,RELAT_1:62,A6,XBOOLE_1:1;
      for x be object st x in D holds (La|D).x = (Lb|D).x
      proof
        let x be object such that A8:x in D;
        reconsider o=x as Ordinal by A8;
        thus (La|D).x = La.o by A8,FUNCT_1:49
        .= Lb.o by A8,ORDINAL1:def 2,A6,A5
        .= (Lb|D).x by A8,FUNCT_1:49;
      end;
      then A9: La|D = Lb|D by A7,FUNCT_1:2;
      A10: D c= B in succ B by A1,A6,ORDINAL1:6,XBOOLE_1:1;
      D in succ A by ORDINAL1:6,A6,ORDINAL1:12;
      hence La.D
       = [:bool union rng (Lb|D), bool union rng (Lb|D):] by A2,A9
      .= Lb.D by A10,A3,ORDINAL1:12;
    end;
    A11:for D be Ordinal holds P[D] from ORDINAL1:sch 2(A4);
    A12:Lb.B = [:bool union rng (Lb|B), bool union rng (Lb|B):]
      by A3,ORDINAL1:6;
    A13: La.A = [:bool union rng (La|A), bool union rng (La|A):]
      by ORDINAL1:6,A2;
    rng (La|A) c= rng (Lb|B)
    proof
      let y be object such that A14:y in rng (La|A);
      consider x be object such that
      A15: x in dom (La|A) & (La|A).x = y by A14,FUNCT_1:def 3;
      A16:dom (La|A)=A & dom (Lb|B)=B by A2,A3,RELAT_1:62,XBOOLE_1:7;
      reconsider x as Ordinal by A15;
      (La|A).x = La.x by A15,FUNCT_1:49
      .= Lb.x by A15,ORDINAL1:def 2,A11
      .=(Lb|B).x by A1,A15,A16,FUNCT_1:49;
      hence thesis by A1,A16,A15,FUNCT_1:def 3;
    end;
    then union rng (La|A) c= union rng (Lb|B) by ZFMISC_1:77;
    then bool union rng (La|A) c= bool union rng (Lb|B) by ZFMISC_1:67;
    hence thesis by A2,A3,A12,A13,ZFMISC_1:96;
end;
