let D, D9, E be non empty set ; 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; 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; for q being FinSequence of D9 holds F .: (p,q) is FinSequence of E
let q be FinSequence of D9; F .: (p,q) is FinSequence of E
A1:
rng (F .: (p,q)) c= rng F
by RELAT_1:26;
( rng p c= D & rng q c= D9 )
by FINSEQ_1:def 4;
then
[:(rng p),(rng q):] c= [:D,D9:]
by ZFMISC_1:96;
then
[:(rng p),(rng q):] c= dom F
by FUNCT_2:def 1;
then A2:
F .: (p,q) is FinSequence
by Th62;
rng F c= E
by RELAT_1:def 19;
then
rng (F .: (p,q)) c= E
by A1;
hence
F .: (p,q) is FinSequence of E
by A2, FINSEQ_1:def 4; verum