let D, D', E be non empty set ; :: thesis: for d1 being Element of D
for d' being Element of D'
for F being Function of [:D,D':],E
for p being FinSequence of D st p = <*d1*> holds
F [:] p,d' = <*(F . d1,d')*>
let d1 be Element of D; :: thesis: for d' being Element of D'
for F being Function of [:D,D':],E
for p being FinSequence of D st p = <*d1*> holds
F [:] p,d' = <*(F . d1,d')*>
let d' be Element of D'; :: thesis: for F being Function of [:D,D':],E
for p being FinSequence of D st p = <*d1*> holds
F [:] p,d' = <*(F . d1,d')*>
let F be Function of [:D,D':],E; :: thesis: for p being FinSequence of D st p = <*d1*> holds
F [:] p,d' = <*(F . d1,d')*>
let p be FinSequence of D; :: thesis: ( p = <*d1*> implies F [:] p,d' = <*(F . d1,d')*> )
assume A1: p = <*d1*> ; :: thesis: F [:] p,d' = <*(F . d1,d')*>
reconsider r = F [:] p,d' as FinSequence of E by Th97;
len p = 1 by A1, FINSEQ_1:56;
then A2: len r = 1 by Th98;
then 1 in Seg (len r) ;
then ( 1 in dom r & p . 1 = d1 ) by A1, FINSEQ_1:57, FINSEQ_1:def 3;
then r . 1 = F . d1,d' by FUNCOP_1:35;
hence F [:] p,d' = <*(F . d1,d')*> by A2, FINSEQ_1:57; :: thesis: verum