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
  F '&' 'not' G is_immediate_constituent_of F => G;
  hence
A1: F '&' 'not' G is_proper_subformula_of F => G by ZF_LANG:61;
  F is_immediate_constituent_of F '&' 'not' G & 'not' G
  is_immediate_constituent_of F '&' 'not' G;
  hence
A2: F is_proper_subformula_of F => G & 'not' G is_proper_subformula_of
  F => G by A1,Th40;
  G is_immediate_constituent_of 'not' G;
  hence thesis by A2,Th40;
end;
