M is ManySortedSubset of M by PBOOLE:def 18;
hence not Bool M is empty by Def1; :: thesis: ( Bool M is functional & Bool M is with_common_domain )
thus Bool M is functional by Def1; :: thesis: Bool M is with_common_domain
let f, g be Function; :: according to CARD_3:def 10 :: thesis: ( not f in Bool M or not g in Bool M or dom f = dom g )
assume that
A1: f in Bool M and
A2: g in Bool M ; :: thesis: dom f = dom g
A3: g is ManySortedSubset of M by A2, Def1;
f is ManySortedSubset of M by A1, Def1;
hence dom f = I by PARTFUN1:def 2
.= dom g by A3, PARTFUN1:def 2 ;
:: thesis: verum