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 = VERUM(A) or H is atomic iff Subformulae H = { H }
proof
  H is atomic implies Subformulae H = { H }
  by Th86;
  hence H = VERUM(A) or H is atomic implies Subformulae H = { H } by Th85;
  assume
A1: Subformulae H = { H };
A2: now
    assume H = 'not' the_argument_of H;
    then
A3: the_argument_of H is_immediate_constituent_of H;
    then the_argument_of H is_proper_subformula_of H by Th53;
    then the_argument_of H is_subformula_of H;
    then
A4: the_argument_of H in Subformulae H by Def22;
    len @(the_argument_of H) <> len @H by A3,Th51;
    hence contradiction by A1,A4,TARSKI:def 1;
  end;
A5: now
    assume H = (the_left_argument_of H) '&' the_right_argument_of H;
    then
A6: the_left_argument_of H is_immediate_constituent_of H;
    then the_left_argument_of H is_proper_subformula_of H by Th53;
    then the_left_argument_of H is_subformula_of H;
    then
A7: the_left_argument_of H in Subformulae H by Def22;
    len @(the_left_argument_of H) <> len @H by A6,Th51;
    hence contradiction by A1,A7,TARSKI:def 1;
  end;
  assume H <> VERUM(A) & not H is atomic;
  then
  H is negative or H is conjunctive or H is universal by QC_LANG1:9;
  then H = 'not' the_argument_of H or H = (the_left_argument_of H) '&'
    the_right_argument_of H or H = All(bound_in H,the_scope_of H)
    by Th3,Th6,QC_LANG1:def 24;
  then
A8: the_scope_of H is_immediate_constituent_of H by A2,A5;
  then the_scope_of H is_proper_subformula_of H by Th53;
  then the_scope_of H is_subformula_of H;
  then
A9: the_scope_of H in Subformulae H by Def22;
  len @(the_scope_of H) <> len @H by A8,Th51;
  hence contradiction by A1,A9,TARSKI:def 1;
end;
