reserve A for QC-alphabet;
reserve sq for FinSequence,
  x,y,z for bound_QC-variable of A,
  p,q,p1,p2,q1 for Element of QC-WFF(A);
reserve s,t for bound_QC-variable of A;
reserve F,G,H,H1 for Element of QC-WFF(A);
reserve x,y,z for bound_QC-variable of A,
  k,n,m for Nat,
  P for ( QC-pred_symbol of k, A),
  V for QC-variable_list of k, A;
reserve L,L9 for FinSequence;

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;
