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 Th88:
  Subformulae 'not' H = Subformulae H \/ { 'not' H }
proof
  thus Subformulae 'not' H c= Subformulae H \/ { 'not' H }
  proof
    let a be object;
    assume a in Subformulae 'not' H;
    then consider F such that
A1: F = a and
A2: F is_subformula_of 'not' H by Def22;
    now
      assume F <> 'not' H;
      then F is_proper_subformula_of 'not' H by A2;
      then F is_subformula_of H by Th66;
      hence a in Subformulae H by A1,Def22;
    end;
    then a in Subformulae H or a in { 'not' H } by A1,TARSKI:def 1;
    hence thesis by XBOOLE_0:def 3;
  end;
  let a be object such that
A3: a in Subformulae H \/ { 'not' H };
A4: now
    assume a in Subformulae H;
    then consider F such that
A5: F = a and
A6: F is_subformula_of H by Def22;
    H is_immediate_constituent_of 'not' H;
    then H is_proper_subformula_of 'not' H by Th53;
    then H is_subformula_of 'not' H;
    then F is_subformula_of 'not' H by A6,Th57;
    hence thesis by A5,Def22;
  end;
  now
    assume a in { 'not' H };
    then a = 'not' H by TARSKI:def 1;
    hence thesis by Def22;
  end;
  hence thesis by A3,A4,XBOOLE_0:def 3;
end;
