let C, D be non empty finite set ; :: thesis: for F being PartFunc of D,REAL
for A being RearrangmentGen of C st F is total & card C = card D holds
dom (Rlor (F,A)) = C
let F be PartFunc of D,REAL; :: thesis: for A being RearrangmentGen of C st F is total & card C = card D holds
dom (Rlor (F,A)) = C
let B be RearrangmentGen of C; :: thesis: ( F is total & card C = card D implies dom (Rlor (F,B)) = C )
set b = Co_Gen B;
A1: ( len (CHI ((Co_Gen B),C)) = len (Co_Gen B) & len ((MIM (FinS (F,D))) (#) (CHI ((Co_Gen B),C))) = min ((len (MIM (FinS (F,D)))),(len (CHI ((Co_Gen B),C)))) ) by RFUNCT_3:def 6, RFUNCT_3:def 7;
assume ( F is total & card C = card D ) ; :: thesis: dom (Rlor (F,B)) = C
then A2: ( len (MIM (FinS (F,D))) = len (CHI ((Co_Gen B),C)) & len (Co_Gen B) = card D ) by Th1, Th11;
thus dom (Rlor (F,B)) c= C ; :: according to XBOOLE_0:def 10 :: thesis: C c= dom (Rlor (F,B))
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in C or x in dom (Rlor (F,B)) )
assume x in C ; :: thesis: x in dom (Rlor (F,B))
then reconsider c = x as Element of C ;
c is_common_for_dom (MIM (FinS (F,D))) (#) (CHI ((Co_Gen B),C)) by RFUNCT_3:32;
hence x in dom (Rlor (F,B)) by A1, A2, RFUNCT_3:28; :: thesis: verum