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 Th44:
  g is nonpositive iff
    for gL st gL in the_LeftOptions_of g
      ex gLR st gLR in the_RightOptions_of gL & gLR is nonpositive
proof
  hereby
    assume g is nonpositive;
    then
A1:   -g is nonnegative;
    let gL;
    assume gL in the_LeftOptions_of g;
    then -gL in the_RightOptions_of -g by Th39;
    then consider gRL such that
A2:   gRL in the_LeftOptions_of -gL & gRL is nonnegative by A1,Th43;
    take gLR = -gRL;
    gLR in the_RightOptions_of -(-gL) & -(-gL) = gL & -gLR = gRL
      by A2,Th39,Th40;
    hence gLR in the_RightOptions_of gL & gLR is nonpositive by A2;
  end;

  assume
A3: for gL st gL in the_LeftOptions_of g
      ex gLR st gLR in the_RightOptions_of gL & gLR is nonpositive;
  now
    let gR;
    assume gR in the_RightOptions_of -g;
    then -gR in the_LeftOptions_of -(-g) & -(-g) = g
      by Th39,Th40;
    then consider gLR such that
A4:   gLR in the_RightOptions_of -gR & gLR is nonpositive by A3;
    take gRL = -gLR;
    gRL in the_LeftOptions_of -(-gR) & -(-gR) = gR
      by A4,Th39,Th40;
    hence gRL in the_LeftOptions_of gR & gRL is nonnegative
      by A4;
  end;
  then -g is nonnegative by Th43;
  hence g is nonpositive;
end;
