theorem
  (H is_immediate_constituent_of G or H is_proper_subformula_of G or H
  is_subformula_of G) & G in Subformulae F implies H in Subformulae F
proof
  assume that
A1: H is_immediate_constituent_of G or H is_proper_subformula_of G or H
  is_subformula_of G and
A2: G in Subformulae F;
  H is_proper_subformula_of G or H is_subformula_of G by A1,Th61;
  then
A3: H is_subformula_of G;
  G is_subformula_of F by A2,Th78;
  then H is_subformula_of F by A3,Th65;
  hence thesis by Def42;
end;
