let D, D', E be non empty set ; :: thesis: for d' being Element of D'
for F being Function of [:D,D':],E
for p being FinSequence of D holds F [:] p,d' is FinSequence of E
let d' be Element of D'; :: thesis: for F being Function of [:D,D':],E
for p being FinSequence of D holds F [:] p,d' is FinSequence of E
let F be Function of [:D,D':],E; :: thesis: for p being FinSequence of D holds F [:] p,d' is FinSequence of E
let p be FinSequence of D; :: thesis: F [:] p,d' is FinSequence of E
( {d'} c= D' & rng p c= D ) by FINSEQ_1:def 4;
then ( [:(rng p),{d'}:] c= [:D,D':] & F [:] p,d' = F * <:p,((dom p) --> d'):> ) by FUNCOP_1:def 4, ZFMISC_1:119;
then ( [:(rng p),{d'}:] c= dom F & rng (F [:] p,d') c= rng F & rng F c= E ) by FUNCT_2:def 1, RELAT_1:45, RELAT_1:def 19;
then ( F [:] p,d' is FinSequence & rng (F [:] p,d') c= E ) by Th82, XBOOLE_1:1;
hence F [:] p,d' is FinSequence of E by FINSEQ_1:def 4; :: thesis: verum