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 Th46:
  g is fuzzy iff
    (ex gL st gL in the_LeftOptions_of g & gL is nonnegative) &
    (ex gR st gR in the_RightOptions_of g & gR is nonpositive)
proof
  hereby
    assume
A1:   g is fuzzy;
    hereby
      consider gL such that
A2:     gL in the_LeftOptions_of g &
          for gLR st gLR in the_RightOptions_of gL holds gLR is non nonpositive
        by A1,Th44;
      take gL;
      thus gL in the_LeftOptions_of g by A2;
      for gLR st gLR in the_RightOptions_of gL
      holds gLR is fuzzy or gLR is positive by A2;
      hence gL is nonnegative by Th45;
    end;
    hereby
      consider gR such that
A3:     gR in the_RightOptions_of g &
          for gRL st gRL in the_LeftOptions_of gR holds gRL is non nonnegative
        by A1,Th43;
      take gR;
      thus gR in the_RightOptions_of g by A3;
      for gRL st gRL in the_LeftOptions_of gR holds
      gRL is fuzzy or gRL is negative by A3;
      hence gR is nonpositive by Th45;
    end;
  end;
  hereby
    assume ex gL st gL in the_LeftOptions_of g & gL is nonnegative;
    then consider gL such that
A4:   gL in the_LeftOptions_of g & gL is nonnegative;
    assume ex gR st gR in the_RightOptions_of g & gR is nonpositive;
    then consider gR such that
A5:   gR in the_RightOptions_of g & gR is nonpositive;

    gL is non fuzzy non negative & gR is non fuzzy non positive by A4,A5;
    then g is non nonpositive & g is non nonnegative by Th45,A4,A5;
    hence g is fuzzy;
  end;
end;
