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
  g is negative iff
    (for gL st gL in the_LeftOptions_of g holds gL is fuzzy or gL is negative)
    & (ex gR st gR in the_RightOptions_of g & gR is nonpositive)
proof
  hereby
    assume
A1:   g is negative;
    hence
A2:   for gL st gL in the_LeftOptions_of g holds
        gL is fuzzy or gL is negative by Th45;
    consider gR such that
A3:   gR in the_RightOptions_of g & gR is non fuzzy non positive
      by Th47,A1,A2;
    take gR;
    thus gR in the_RightOptions_of g & gR is nonpositive by A3;
  end;
  hereby
    assume
     for gL st gL in the_LeftOptions_of g holds gL is fuzzy or gL is negative;
    then
A4:   g is nonpositive by Th45;
    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;
    gR is non fuzzy non positive by A5;
    then not g is zero by Th47,A5;
    hence g is negative by A4;
  end;
end;
