let D, D9, E be non empty set ; :: thesis: for F being Function of [:D,D9:],E
for p being FinSequence of D
for q being FinSequence of D9 holds F .: p,q is FinSequence of E
let F be Function of [:D,D9:],E; :: thesis: for p being FinSequence of D
for q being FinSequence of D9 holds F .: p,q is FinSequence of E
let p be FinSequence of D; :: thesis: for q being FinSequence of D9 holds F .: p,q is FinSequence of E
let q be FinSequence of D9; :: thesis: F .: p,q is FinSequence of E
F .: p,q = F * <:p,q:> by FUNCOP_1:def 3;
then A1: rng (F .: p,q) c= rng F by RELAT_1:45;
( rng p c= D & rng q c= D9 ) by FINSEQ_1:def 4;
then [:(rng p),(rng q):] c= [:D,D9:] by ZFMISC_1:119;
then [:(rng p),(rng q):] c= dom F by FUNCT_2:def 1;
then A2: F .: p,q is FinSequence by Th78;
rng F c= E by RELAT_1:def 19;
then rng (F .: p,q) c= E by A1, XBOOLE_1:1;
hence F .: p,q is FinSequence of E by A2, FINSEQ_1:def 4; :: thesis: verum