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 Th80:
  H is_subformula_of P!V iff H = P!V
proof
  thus H is_subformula_of P!V implies H = P!V
  proof
    assume
A1: H is_subformula_of P!V;
    assume H <> P!V;
    then H is_proper_subformula_of P!V by A1;
    then ex F st F is_immediate_constituent_of P!V by Th55;
    hence contradiction by Th42;
  end;
  thus thesis;
end;
