let A be QC-alphabet ; :: thesis: for p being Element of QC-WFF A
for V being non empty Subset of (QC-variables A) st p is disjunctive holds
Vars (p,V) = (Vars ((the_left_disjunct_of p),V)) \/ (Vars ((the_right_disjunct_of p),V))
let p be Element of QC-WFF A; :: thesis: for V being non empty Subset of (QC-variables A) st p is disjunctive holds
Vars (p,V) = (Vars ((the_left_disjunct_of p),V)) \/ (Vars ((the_right_disjunct_of p),V))
let V be non empty Subset of (QC-variables A); :: thesis: ( p is disjunctive implies Vars (p,V) = (Vars ((the_left_disjunct_of p),V)) \/ (Vars ((the_right_disjunct_of p),V)) )
set p1 = the_left_disjunct_of p;
set p2 = the_right_disjunct_of p;
assume p is disjunctive ; :: thesis: Vars (p,V) = (Vars ((the_left_disjunct_of p),V)) \/ (Vars ((the_right_disjunct_of p),V))
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;
hence Vars (p,V) = Vars ((('not' (the_left_disjunct_of p)) '&' ('not' (the_right_disjunct_of p))),V) by Th39
.= (Vars (('not' (the_left_disjunct_of p)),V)) \/ (Vars (('not' (the_right_disjunct_of p)),V)) by Th42
.= (Vars ((the_left_disjunct_of p),V)) \/ (Vars (('not' (the_right_disjunct_of p)),V)) by Th39
.= (Vars ((the_left_disjunct_of p),V)) \/ (Vars ((the_right_disjunct_of p),V)) by Th39 ;
:: thesis: verum