let D, D9, E be non empty set ; :: thesis: for d being Element of D
for F being Function of [:D,D9:],E
for p being FinSequence of D9 holds F [;] d,p is FinSequence of E
let d be Element of D; :: thesis: for F being Function of [:D,D9:],E
for p being FinSequence of D9 holds F [;] d,p is FinSequence of E
let F be Function of [:D,D9:],E; :: thesis: for p being FinSequence of D9 holds F [;] d,p is FinSequence of E
let p be FinSequence of D9; :: thesis: F [;] d,p is FinSequence of E
F [;] d,p = F * <:((dom p) --> d),p:> by FUNCOP_1:def 5;
then A1: rng (F [;] d,p) c= rng F by RELAT_1:45;
rng p c= D9 by FINSEQ_1:def 4;
then [:{d},(rng p):] c= [:D,D9:] by ZFMISC_1:119;
then [:{d},(rng p):] c= dom F by FUNCT_2:def 1;
then A2: F [;] d,p is FinSequence by Th80;
rng F c= E by RELAT_1:def 19;
then rng (F [;] d,p) c= E by A1, XBOOLE_1:1;
hence F [;] d,p is FinSequence of E by A2, FINSEQ_1:def 4; :: thesis: verum