let A be QC-alphabet ; :: thesis: for F, G being Element of QC-WFF A holds
( 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 )
let F, G be Element of QC-WFF A; :: thesis: ( 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 )
thus A1: F '&' ('not' G) is_proper_subformula_of F => G by Th66; :: thesis: ( F is_proper_subformula_of F => G & 'not' G is_proper_subformula_of F => G & G is_proper_subformula_of F => G )
( 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; :: thesis: G is_proper_subformula_of F => G
G is_proper_subformula_of 'not' G by Th66;
hence G is_proper_subformula_of F => G by A2, Th56; :: thesis: verum