reserve x,y,y1,y2,z,e,s for set;
reserve alpha,beta,gamma for Ordinal;
reserve n,m,k for Nat;

theorem Th1:
 for z being object holds
  z in ConwayDay(alpha) iff ex w being strict left-right st z = w &
    for x st x in (the LeftOptions of w) \/ (the RightOptions of w)
      ex beta st beta in alpha & x in ConwayDay(beta)
proof let z be object;
  consider f being Sequence such that
A1: alpha in dom f & f.alpha = ConwayDay(alpha) & ConwayIteration[f]
    by Def2;
  hereby
    assume z in ConwayDay(alpha);
    then z in ConwayNextDay(f|alpha) by A1;
    then consider x,y being Subset of union(rng (f|alpha)) such that
A2:   z = left-right(#x,y#);
    reconsider w = z as strict left-right by A2;
    take w;
    thus z = w;
    let e;
    assume e in (the LeftOptions of w) \/ (the RightOptions of w);
    then e in x or e in y by A2,XBOOLE_0:def 3;
    then consider r being set such that
A3:   e in r & r in rng (f|alpha) by TARSKI:def 4;
    ex beta being object st beta in dom (f|alpha) & r = (f|alpha).beta
      by A3,FUNCT_1:def 3;
    then consider beta such that
A4:   beta in dom (f|alpha) & r = (f|alpha).beta;
    take beta;
    dom (f|alpha) c= alpha by RELAT_1:58;
    hence beta in alpha by A4;
    dom (f|alpha) c= dom f by RELAT_1:60;
    then f.beta = ConwayDay(beta) by A1,Def2,A4;
    hence e in ConwayDay(beta) by A3,A4,FUNCT_1:47;
  end;

  hereby
    assume ex w being strict left-right st z = w &
      for x st x in (the LeftOptions of w) \/ (the RightOptions of w)
        ex beta st beta in alpha & x in ConwayDay(beta);
    then consider w being strict left-right such that
A5:   w = z & for x st x in (the LeftOptions of w) \/ (the RightOptions of w)
        ex beta st beta in alpha & x in ConwayDay(beta);

    the LeftOptions of w is Subset of union(rng (f|alpha))
      & the RightOptions of w is Subset of union(rng (f|alpha))
    proof
      (the LeftOptions of w) \/ (the RightOptions of w) c= union(rng (f|alpha))
      proof
        let e be object;
        assume e in (the LeftOptions of w) \/ (the RightOptions of w);
        then consider beta such that
A6:       beta in alpha & e in ConwayDay(beta) by A5;

A7:     alpha c= dom f by A1,ORDINAL1:def 2;
        then f.beta = ConwayDay(beta) by Def2,A1,A6;
        then ConwayDay(beta) c= union(rng (f|alpha))
          by A6,A7,FUNCT_1:50,ZFMISC_1:74;
        hence e in union(rng (f|alpha)) by A6;
      end;
      hence thesis by XBOOLE_1:11;
    end;
    then w in ConwayNextDay(f|alpha);
    hence z in ConwayDay(alpha) by A1,A5;
  end;
end;
