let D, D9, E be non empty set ; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: for p being FinSequence of D9 st p = <*d19*> holds
F [;] (d,p) = <*(F . (d,d19))*>
let p be FinSequence of D9; :: thesis: ( p = <*d19*> implies F [;] (d,p) = <*(F . (d,d19))*> )
assume A1: p = <*d19*> ; :: thesis: F [;] (d,p) = <*(F . (d,d19))*>
A2: p . 1 = d19 by A1;
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; :: thesis: verum