theorem Th32:
  g1 in the_Tree_of g2 implies the_proper_Tree_of g1 c= the_proper_Tree_of g2
proof
  assume
A1: g1 in the_Tree_of g2;
  then
A2: the_Tree_of g1 c= the_Tree_of g2 by Th31;
    let x be object;
    assume
A3:   x in the_proper_Tree_of g1;
    then
A4:   x in the_Tree_of g1 & x <> g1 by ZFMISC_1:56;
    assume
A5:   not x in the_proper_Tree_of g2;
    then
A6:   x = g2 by A2,A4,ZFMISC_1:56;
    per cases by Th28,A1;
      suppose g1 = g2;
        hence contradiction by A3,A5;
      end;
      suppose
A7:     ConwayRank(g1) in ConwayRank(g2);
        reconsider g = x as ConwayGame by A3;
        ConwayRank(g) in ConwayRank(g2) by A7,Th29,A4,ORDINAL1:12;
        hence contradiction by A6;
      end;
end;
