theorem Th33:
  the_Options_of g c= the_proper_Tree_of g
proof
  let x be object such that
A1: x in the_Options_of g;
  reconsider gO = x as ConwayGame by A1,Th17;
  gO in the_Tree_of gO by Th24;
  then gO in the_Tree_of g by A1,Th27;
  then gO = g or gO in the_proper_Tree_of g by ZFMISC_1:56;
  hence thesis by A1,Th16;
end;
