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
  (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,Th53;
  then
A3: H is_subformula_of G;
  G is_subformula_of F by A2,Th82;
  then H is_subformula_of F by A3,Th57;
  hence thesis by Def22;
end;
