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 Th89:
  Subformulae (H '&' F) = Subformulae H \/ Subformulae F \/ { H '&' F }
proof
  thus Subformulae H '&' F c= Subformulae H \/ Subformulae F \/ { H '&' F }
  proof
    let a be object;
    assume a in Subformulae H '&' F;
    then consider G such that
A1: G = a and
A2: G is_subformula_of H '&' F by Def22;
    now
      assume G <> H '&' F;
      then G is_proper_subformula_of H '&' F by A2;
      then G is_subformula_of H or G is_subformula_of F by Th69;
      then a in Subformulae H or a in Subformulae F by A1,Def22;
      hence a in Subformulae H \/ Subformulae F by XBOOLE_0:def 3;
    end;
    then a in Subformulae H \/ Subformulae F or a in { H '&' F } by A1,
TARSKI:def 1;
    hence thesis by XBOOLE_0:def 3;
  end;
  let a be object such that
A3: a in Subformulae H \/ Subformulae F \/ { H '&' F };
A4: now
    assume a in { H '&' F };
    then a = H '&' F by TARSKI:def 1;
    hence thesis by Def22;
  end;
A5: now
    assume a in Subformulae F;
    then consider G such that
A6: G = a and
A7: G is_subformula_of F by Def22;
    F is_immediate_constituent_of H '&' F;
    then F is_proper_subformula_of H '&' F by Th53;
    then F is_subformula_of H '&' F;
    then G is_subformula_of H '&' F by A7,Th57;
    hence thesis by A6,Def22;
  end;
A8: now
    assume a in Subformulae H;
    then consider G such that
A9: G = a and
A10: G is_subformula_of H by Def22;
    H is_immediate_constituent_of H '&' F;
    then H is_proper_subformula_of H '&' F by Th53;
    then H is_subformula_of H '&' F;
    then G is_subformula_of H '&' F by A10,Th57;
    hence thesis by A9,Def22;
  end;
  a in Subformulae H \/ Subformulae F implies a in Subformulae H or a in
  Subformulae F by XBOOLE_0:def 3;
  hence thesis by A3,A8,A5,A4,XBOOLE_0:def 3;
end;
