let p be QC-formula; :: thesis: ( p is conditional implies still_not-bound_in p = (still_not-bound_in (the_antecedent_of p)) \/ (still_not-bound_in (the_consequent_of p)) )
assume p is conditional ; :: thesis: still_not-bound_in p = (still_not-bound_in (the_antecedent_of p)) \/ (still_not-bound_in (the_consequent_of p))
then A1: p = (the_antecedent_of p) => (the_consequent_of p) by QC_LANG2:54;
set p1 = the_antecedent_of p;
set p2 = the_consequent_of p;
p = 'not' ((the_antecedent_of p) '&' ('not' (the_consequent_of p))) by A1, QC_LANG2:def 2;
then still_not-bound_in p = still_not-bound_in ((the_antecedent_of p) '&' ('not' (the_consequent_of p))) by Th11
.= (still_not-bound_in (the_antecedent_of p)) \/ (still_not-bound_in ('not' (the_consequent_of p))) by Th14
.= (still_not-bound_in (the_antecedent_of p)) \/ (still_not-bound_in (the_consequent_of p)) by Th11 ;
hence still_not-bound_in p = (still_not-bound_in (the_antecedent_of p)) \/ (still_not-bound_in (the_consequent_of p)) ; :: thesis: verum