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