theorem Th81:
  H is_subformula_of FALSUM(A) iff H = FALSUM(A) or H = VERUM(A)
proof
  thus H is_subformula_of FALSUM(A) implies H = FALSUM(A) or H = VERUM(A)
  proof
    assume H is_subformula_of FALSUM(A) & H <> FALSUM(A);
    then H is_proper_subformula_of FALSUM(A);
    then H is_subformula_of VERUM(A) by Th66;
    hence thesis by Th79;
  end;
  VERUM(A) is_immediate_constituent_of FALSUM(A);
  then VERUM(A) is_proper_subformula_of FALSUM(A) by Th53;
  hence thesis;
end;
