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