reserve x,y,y1,y2,z,e,s for set;
reserve alpha,beta,gamma for Ordinal;
reserve n,m,k for Nat;
reserve g,g0,g1,g2,gO,gL,gR,gLL,gLR,gRL,gRR for ConwayGame;

theorem Th43:
  g is nonnegative iff
    for gR st gR in the_RightOptions_of g
      ex gRL st gRL in the_LeftOptions_of gR & gRL is nonnegative
proof
  defpred S[set] means
    for g st g in $1 for gR st gR in the_RightOptions_of g
      ex gRL st gRL in the_LeftOptions_of gR & gRL in $1;
  A1:
    S[s] implies S[s /\ ConwayDay(alpha)]
  proof
    assume
  A2: S[s];
    let g;
    assume g in s /\ ConwayDay(alpha);
    then
  A3: g in s & g in ConwayDay(alpha) by XBOOLE_0:def 4;
    let gR;
    assume
  A4: gR in the_RightOptions_of g;
    then consider gRL such that
  A5: gRL in the_LeftOptions_of gR & gRL in s by A2,A3;
    take gRL;
    gR in ConwayDay(alpha) by Th11,A3,A4;
    then gRL in ConwayDay(alpha) by Th11,A5;
    hence thesis by A5,XBOOLE_0:def 4;
  end;

  hereby
    assume g is nonnegative;
    then consider s such that
A6:   g in s & S[s];
    let gR;
    assume gR in the_RightOptions_of g;
    then consider gRL such that
A7:   gRL in the_LeftOptions_of gR & gRL in s by A6;
    take gRL;
    thus gRL in the_LeftOptions_of gR by A7;
    thus gRL is nonnegative
    by A6,A7;
  end;

  hereby
    assume
A8:   for gR st gR in the_RightOptions_of g
        ex gRL st gRL in the_LeftOptions_of gR & gRL is nonnegative;

    consider alpha such that
A9:   g in ConwayDay(alpha) by Def3;

    now
      set ss = { s1 where s1 is Subset of ConwayDay(alpha) : S[s1] };
      take s = union ss;

A10:   S[s]
      proof
        let g1;
        assume g1 in s;
        then consider s2 being set such that
A11:       g1 in s2 & s2 in ss by TARSKI:def 4;
        consider s1 being Subset of ConwayDay(alpha) such that
A12:       s1 = s2 & S[s1] by A11;
        let gR;
        assume gR in the_RightOptions_of g1;
        then consider gRL such that
A13:       gRL in the_LeftOptions_of gR & gRL in s1 by A11,A12;
        take gRL;
        s2 c= s by A11,ZFMISC_1:74;
        hence thesis by A12,A13;
      end;

      thus g in s
      proof
        now
          let x;
          assume x in ss;
          then ex s1 being Subset of ConwayDay(alpha) st x = s1 & S[s1];
          hence x c= ConwayDay(alpha);
        end;
        then
A14:       s c= ConwayDay(alpha) by ZFMISC_1:76;

        {g} c= ConwayDay(alpha) by A9,ZFMISC_1:31;
        then reconsider sg = s \/ {g} as Subset of ConwayDay(alpha)
          by A14,XBOOLE_1:8;

        S[sg]
        proof
          let g1 such that
A15:         g1 in sg;
          let gR such that
A16:        gR in the_RightOptions_of g1;
          per cases by A15,XBOOLE_0:def 3;
            suppose g1 in s;
              then consider gRL such that
A17:            gRL in the_LeftOptions_of gR & gRL in s by A10,A16;
              take gRL;
              thus gRL in the_LeftOptions_of gR by A17;
              s c= sg by XBOOLE_1:7;
              hence gRL in sg by A17;
            end;
            suppose g1 in {g};
              then g1 = g by TARSKI:def 1;
              then consider gRL such that
A18:            gRL in the_LeftOptions_of gR & gRL is nonnegative by A8,A16;
              consider s0 being set such that
A19:            gRL in s0 & S[s0] by A18;
              take gRL;
              thus gRL in the_LeftOptions_of gR by A18;

              reconsider s1 = s0 /\ ConwayDay(alpha)
                as Subset of ConwayDay(alpha) by XBOOLE_1:17;

              S[s1] by A19,A1;
              then s1 in ss;
              then
A20:            s1 c= s by ZFMISC_1:74;

              gR in ConwayDay(alpha) by A15,A16,Th11;
              then gRL in ConwayDay(alpha) by A18,Th11;
              then gRL in s1 by A19,XBOOLE_0:def 4;
              hence gRL in sg by A20,XBOOLE_0:def 3;
            end;
        end;
        then
A21:       sg in ss;

        g in {g} by TARSKI:def 1;
        then g in sg by XBOOLE_0:def 3;
        hence g in s by A21,TARSKI:def 4;
      end;
      thus S[s] by A10;
    end;
    hence g is nonnegative;
  end;
end;
