let D, D9, E be non empty set ; :: thesis: for d being Element of D
for d19, d29, d39 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D9 st p = <*d19,d29,d39*> holds
F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*>
let d be Element of D; :: thesis: for d19, d29, d39 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D9 st p = <*d19,d29,d39*> holds
F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*>
let d19, d29, d39 be Element of D9; :: thesis: for F being Function of [:D,D9:],E
for p being FinSequence of D9 st p = <*d19,d29,d39*> holds
F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*>
let F be Function of [:D,D9:],E; :: thesis: for p being FinSequence of D9 st p = <*d19,d29,d39*> holds
F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*>
let p be FinSequence of D9; :: thesis: ( p = <*d19,d29,d39*> implies F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*> )
assume A1: p = <*d19,d29,d39*> ; :: thesis: F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*>
A2: p . 2 = d29 by A1, FINSEQ_1:45;
reconsider r = F [;] (d,p) as FinSequence of E by Th75;
len p = 3 by A1, FINSEQ_1:45;
then A3: len r = 3 by Th76;
then 2 in Seg (len r) ;
then 2 in dom r by FINSEQ_1:def 3;
then A4: r . 2 = F . (d,d29) by A2, FUNCOP_1:32;
A5: p . 3 = d39 by A1, FINSEQ_1:45;
A6: p . 1 = d19 by A1, FINSEQ_1:45;
3 in Seg (len r) by A3;
then 3 in dom r by FINSEQ_1:def 3;
then A7: r . 3 = F . (d,d39) by A5, FUNCOP_1:32;
1 in Seg (len r) by A3;
then 1 in dom r by FINSEQ_1:def 3;
then r . 1 = F . (d,d19) by A6, FUNCOP_1:32;
hence F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*> by A3, A4, A7, FINSEQ_1:45; :: thesis: verum