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
F [:] p,d' = F * <:p,((dom p) --> d'):> by FUNCOP_1:def 4;
then A1: rng (F [:] p,d') c= rng F by RELAT_1:45;
rng p c= D by FINSEQ_1:def 4;
then [:(rng p),{d'}:] c= [:D,D':] by ZFMISC_1:119;
then [:(rng p),{d'}:] c= dom F by FUNCT_2:def 1;
then A2: F [:] p,d' is FinSequence by Th82;
rng F c= E by RELAT_1:def 19;
then rng (F [:] p,d') c= E by A1, XBOOLE_1:1;
hence F [:] p,d' is FinSequence of E by A2, FINSEQ_1:def 4; :: thesis: verum