let D, D9, E be non empty set ; :: thesis: for d9 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D holds F [:] p,d9 is FinSequence of E
let d9 be Element of D9; :: thesis: for F being Function of [:D,D9:],E
for p being FinSequence of D holds F [:] p,d9 is FinSequence of E
let F be Function of [:D,D9:],E; :: thesis: for p being FinSequence of D holds F [:] p,d9 is FinSequence of E
let p be FinSequence of D; :: thesis: F [:] p,d9 is FinSequence of E
F [:] p,d9 = F * <:p,((dom p) --> d9):> by FUNCOP_1:def 4;
then A1: rng (F [:] p,d9) c= rng F by RELAT_1:45;
rng p c= D by FINSEQ_1:def 4;
then [:(rng p),{d9}:] c= [:D,D9:] by ZFMISC_1:119;
then [:(rng p),{d9}:] c= dom F by FUNCT_2:def 1;
then A2: F [:] p,d9 is FinSequence by Th82;
rng F c= E by RELAT_1:def 19;
then rng (F [:] p,d9) c= E by A1, XBOOLE_1:1;
hence F [:] p,d9 is FinSequence of E by A2, FINSEQ_1:def 4; :: thesis: verum