let D, D9, E be non empty set ; for d being Element of D
for d19 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D9 st p = <*d19*> holds
F [;] (d,p) = <*(F . (d,d19))*>
let d be Element of D; for d19 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D9 st p = <*d19*> holds
F [;] (d,p) = <*(F . (d,d19))*>
let d19 be Element of D9; for F being Function of [:D,D9:],E
for p being FinSequence of D9 st p = <*d19*> holds
F [;] (d,p) = <*(F . (d,d19))*>
let F be Function of [:D,D9:],E; for p being FinSequence of D9 st p = <*d19*> holds
F [;] (d,p) = <*(F . (d,d19))*>
let p be FinSequence of D9; ( p = <*d19*> implies F [;] (d,p) = <*(F . (d,d19))*> )
assume A1:
p = <*d19*>
; F [;] (d,p) = <*(F . (d,d19))*>
A2:
p . 1 = d19
by A1, FINSEQ_1:40;
reconsider r = F [;] (d,p) as FinSequence of E by Th75;
len p = 1
by A1, FINSEQ_1:39;
then A3:
len r = 1
by Th76;
then
1 in Seg (len r)
;
then
1 in dom r
by FINSEQ_1:def 3;
then
r . 1 = F . (d,d19)
by A2, FUNCOP_1:32;
hence
F [;] (d,p) = <*(F . (d,d19))*>
by A3, FINSEQ_1:40; verum