let A be QC-alphabet ; :: thesis: for p being QC-formula of A st p is disjunctive holds
still_not-bound_in p = (still_not-bound_in (the_left_disjunct_of p)) \/ (still_not-bound_in (the_right_disjunct_of p))
let p be QC-formula of A; :: thesis: ( p is disjunctive implies still_not-bound_in p = (still_not-bound_in (the_left_disjunct_of p)) \/ (still_not-bound_in (the_right_disjunct_of p)) )
set p1 = the_left_disjunct_of p;
set p2 = the_right_disjunct_of p;
assume p is disjunctive ; :: thesis: still_not-bound_in p = (still_not-bound_in (the_left_disjunct_of p)) \/ (still_not-bound_in (the_right_disjunct_of p))
then p = (the_left_disjunct_of p) 'or' (the_right_disjunct_of p) by QC_LANG2:37;
then p = 'not' (('not' (the_left_disjunct_of p)) '&' ('not' (the_right_disjunct_of p))) by QC_LANG2:def 3;
then still_not-bound_in p = still_not-bound_in (('not' (the_left_disjunct_of p)) '&' ('not' (the_right_disjunct_of p))) by Th7
.= (still_not-bound_in ('not' (the_left_disjunct_of p))) \/ (still_not-bound_in ('not' (the_right_disjunct_of p))) by Th10
.= (still_not-bound_in (the_left_disjunct_of p)) \/ (still_not-bound_in ('not' (the_right_disjunct_of p))) by Th7
.= (still_not-bound_in (the_left_disjunct_of p)) \/ (still_not-bound_in (the_right_disjunct_of p)) by Th7 ;
hence still_not-bound_in p = (still_not-bound_in (the_left_disjunct_of p)) \/ (still_not-bound_in (the_right_disjunct_of p)) ; :: thesis: verum