theorem Th37:
  s in G1-entry_points_in_subformula_tree_of G2 implies t1^s is
  Entry_Point_in_Subformula_Tree of G2
proof
  (tree_of_subformulae(F)).t1 = G1 by Def5;
  then
A1: t1 in F-entry_points_in_subformula_tree_of G1 by Def3;
  assume s in G1-entry_points_in_subformula_tree_of G2;
  then
A2: t1^s in F-entry_points_in_subformula_tree_of G2 by A1,Th27;
  F-entry_points_in_subformula_tree_of G2 c= dom tree_of_subformulae(F) by
TREES_1:def 11;
  then reconsider t9 = t1^s as Element of dom tree_of_subformulae(F) by A2;
  (tree_of_subformulae(F)).t9 = G2 by A2,Def3;
  hence thesis by Def5;
end;
