let D, D', E be non empty set ; :: thesis: for d being Element of D
for d1' being Element of D'
for F being Function of [:D,D':],E
for p being FinSequence of D' st p = <*d1'*> holds
F [;] d,p = <*(F . d,d1')*>
let d be Element of D; :: thesis: for d1' being Element of D'
for F being Function of [:D,D':],E
for p being FinSequence of D' st p = <*d1'*> holds
F [;] d,p = <*(F . d,d1')*>
let d1' be Element of D'; :: thesis: for F being Function of [:D,D':],E
for p being FinSequence of D' st p = <*d1'*> holds
F [;] d,p = <*(F . d,d1')*>
let F be Function of [:D,D':],E; :: thesis: for p being FinSequence of D' st p = <*d1'*> holds
F [;] d,p = <*(F . d,d1')*>
let p be FinSequence of D'; :: thesis: ( p = <*d1'*> implies F [;] d,p = <*(F . d,d1')*> )
assume A1: p = <*d1'*> ; :: thesis: F [;] d,p = <*(F . d,d1')*>
reconsider r = F [;] d,p as FinSequence of E by Th91;
len p = 1 by A1, FINSEQ_1:56;
then A2: len r = 1 by Th92;
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 . d,d1' by FUNCOP_1:42;
hence F [;] d,p = <*(F . d,d1')*> by A2, FINSEQ_1:57; :: thesis: verum