reserve k,m,n for Element of NAT,
  a,X,Y for set,
  D,D1,D2 for non empty set;
reserve p,q for FinSequence of NAT;
reserve x,y,z,t for Variable;
reserve F,F1,G,G1,H,H1 for ZF-formula;
reserve sq,sq9 for FinSequence;
reserve L,L9 for FinSequence;
reserve j,j1 for Element of NAT;

theorem
  H is atomic iff Subformulae H = { H }
proof
  thus H is atomic implies Subformulae H = { H }
  proof
    assume H is atomic;
    then H is being_equality or H is being_membership;
    then H = (Var1 H) '=' Var2 H or H = (Var1 H) 'in' Var2 H by Th36,Th37;
    hence thesis by Th80,Th81;
  end;
  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 Th61;
    then the_argument_of H is_subformula_of H;
    then
A4: the_argument_of H in Subformulae H by Def42;
    len(the_argument_of H) <> len H by A3,Th60;
    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 Th61;
    then the_left_argument_of H is_subformula_of H;
    then
A7: the_left_argument_of H in Subformulae H by Def42;
    len(the_left_argument_of H) <> len H by A6,Th60;
    hence contradiction by A1,A7,TARSKI:def 1;
  end;
  assume not H is atomic;
  then H is negative or H is conjunctive or H is universal by Th10;
  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 Def30,Th40
,Th44;
  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 Th61;
  then the_scope_of H is_subformula_of H;
  then
A9: the_scope_of H in Subformulae H by Def42;
  len(the_scope_of H) <> len H by A8,Th60;
  hence contradiction by A1,A9,TARSKI:def 1;
end;
