let D, D9, E be non empty set ; 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; 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; 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; 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; ( p = <*d19,d29,d39*> implies F [;] (d,p) = <*(F . (d,d19)),(F . (d,d29)),(F . (d,d39))*> )
assume A1:
p = <*d19,d29,d39*>
; 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; verum