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