theorem Th35:
  g1 in the_proper_Tree_of g2 implies the_Tree_of g1 c= the_proper_Tree_of g2
proof
  assume
A1:  g1 in the_proper_Tree_of g2;
  then
A2: g1 in the_Tree_of g2 & g1 <> g2 by ZFMISC_1:56;
A3:
  the_Tree_of g1 c= the_Tree_of g2 by Th31,A1;
  not g2 in the_Tree_of g1
  proof
A4: ConwayRank(g1) in ConwayRank(g2) by A2,Th28;
    assume g2 in the_Tree_of g1;
    then ConwayRank(g2) c= ConwayRank(g1) by Th29;
    then ConwayRank(g1) in ConwayRank(g1) by A4;
    hence contradiction;
  end;
  hence thesis by A3,ZFMISC_1:34;
end;
