theorem Th82:
  Subformulae 'not' H = Subformulae H \/ { 'not' H }
proof
  now
    let a be object;
A1: now
      assume a in { 'not' H };
      then
A2:   a = 'not' H by TARSKI:def 1;
      'not' H is_subformula_of 'not' H by Th59;
      hence a in Subformulae 'not' H by A2,Def42;
    end;
    thus a in Subformulae 'not' H implies a in Subformulae H \/ { 'not' H }
    proof
      assume a in Subformulae 'not' H;
      then consider F such that
A3:   F = a and
A4:   F is_subformula_of 'not' H by Def42;
      now
        assume F <> 'not' H;
        then F is_proper_subformula_of 'not' H by A4;
        then F is_subformula_of H by Th69;
        hence a in Subformulae H by A3,Def42;
      end;
      then a in Subformulae H or a in { 'not' H } by A3,TARSKI:def 1;
      hence thesis by XBOOLE_0:def 3;
    end;
A5: now
      assume a in Subformulae H;
      then consider F such that
A6:   F = a and
A7:   F is_subformula_of H by Def42;
      H is_immediate_constituent_of 'not' H;
      then H is_proper_subformula_of 'not' H by Th61;
      then H is_subformula_of 'not' H;
      then F is_subformula_of 'not' H by A7,Th65;
      hence a in Subformulae 'not' H by A6,Def42;
    end;
    assume a in Subformulae H \/ { 'not' H };
    hence a in Subformulae 'not' H by A5,A1,XBOOLE_0:def 3;
  end;
  hence thesis by TARSKI:2;
end;
