let D, D9, E be non empty set ; for d1 being Element of D
for d19 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D
for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds
F .: (p,q) = <*(F . (d1,d19))*>
let d1 be Element of D; for d19 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D
for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds
F .: (p,q) = <*(F . (d1,d19))*>
let d19 be Element of D9; for F being Function of [:D,D9:],E
for p being FinSequence of D
for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds
F .: (p,q) = <*(F . (d1,d19))*>
let F be Function of [:D,D9:],E; for p being FinSequence of D
for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds
F .: (p,q) = <*(F . (d1,d19))*>
let p be FinSequence of D; for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds
F .: (p,q) = <*(F . (d1,d19))*>
let q be FinSequence of D9; ( p = <*d1*> & q = <*d19*> implies F .: (p,q) = <*(F . (d1,d19))*> )
assume A1:
( p = <*d1*> & q = <*d19*> )
; F .: (p,q) = <*(F . (d1,d19))*>
A2:
( p . 1 = d1 & q . 1 = d19 )
by A1, FINSEQ_1:40;
reconsider r = F .: (p,q) as FinSequence of E by Th68;
( len p = 1 & len q = 1 )
by A1, FINSEQ_1:39;
then A3:
len r = 1
by Th70;
then
1 in Seg (len r)
;
then
1 in dom r
by FINSEQ_1:def 3;
then
r . 1 = F . (d1,d19)
by A2, FUNCOP_1:22;
hence
F .: (p,q) = <*(F . (d1,d19))*>
by A3, FINSEQ_1:40; verum