theorem
  'not' F '&' 'not' G is_proper_subformula_of F 'or' G & 'not' F
is_proper_subformula_of F 'or' G & 'not' G is_proper_subformula_of F 'or' G & F
  is_proper_subformula_of F 'or' G & G is_proper_subformula_of F 'or' G
proof
  thus
A1: 'not' F '&' 'not' G is_proper_subformula_of F 'or' G by Th66;
  'not' F is_proper_subformula_of 'not' F '&' 'not' G & 'not' G
  is_proper_subformula_of 'not' F '&' 'not' G by Th69;
  hence
A2: 'not' F is_proper_subformula_of F 'or' G & 'not' G
  is_proper_subformula_of F 'or' G by A1,Th56;
  G is_proper_subformula_of 'not' G & F is_proper_subformula_of 'not' F by Th66
;
  hence thesis by A2,Th56;
end;
