theorem Th45:
  G in Subformulae H iff G is_subformula_of H
proof
  G in Subformulae H implies G is_subformula_of H
  proof
    assume G in Subformulae H;
    then ex F st F = G & F is_subformula_of H by Def24;
    hence thesis;
  end;
  hence thesis by Def24;
end;
