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
  Subformulae (F <=> G) = Subformulae F \/ Subformulae G \/ { 'not' G, F
  '&' 'not' G, F => G, 'not' F, G '&' 'not' F, G => F, F <=> G }
proof
  thus Subformulae (F <=> G) = Subformulae(F => G) \/ Subformulae(G => F) \/ {
  F <=> G} by Th89
    .= Subformulae F \/ Subformulae G \/ { 'not' G, F '&' 'not' G, F => G }
  \/ Subformulae(G => F) \/ {F <=> G} by Th91
    .= Subformulae F \/ Subformulae G \/ { 'not' G, F '&' 'not' G, F => G }
\/ (Subformulae F \/ Subformulae G \/ { 'not' F, G '&' 'not' F, G => F }) \/ {F
  <=> G} by Th91
    .= Subformulae F \/ Subformulae G \/ ((Subformulae F \/ Subformulae G \/
{ 'not' G, F '&' 'not' G, F => G }) \/ { 'not' F, G '&' 'not' F, G => F }) \/ {
  F <=> G} by XBOOLE_1:4
    .= Subformulae F \/ Subformulae G \/ (Subformulae F \/ Subformulae G \/
({ 'not' G, F '&' 'not' G, F => G } \/ { 'not' F, G '&' 'not' F, G => F })) \/
  {F <=> G} by XBOOLE_1:4
    .= Subformulae F \/ Subformulae G \/ (Subformulae F \/ Subformulae G) \/
({ 'not' G, F '&' 'not' G, F => G } \/ { 'not' F, G '&' 'not' F, G => F }) \/ {
  F <=> G} by XBOOLE_1:4
    .= Subformulae F \/ Subformulae G \/ { 'not' G, F '&' 'not' G, F => G,
  'not' F, G '&' 'not' F, G => F } \/ {F <=> G} by ENUMSET1:13
    .= Subformulae F \/ Subformulae G \/ ({ 'not' G, F '&' 'not' G, F => G,
  'not' F, G '&' 'not' F, G => F } \/ {F <=> G}) by XBOOLE_1:4
    .= Subformulae F \/ Subformulae G \/ { 'not' G, F '&' 'not' G, F => G,
  'not' F, G '&' 'not' F, G => F, F <=> G } by ENUMSET1:21;
end;
