let D, D9, E be non empty set ; for d1, d2 being Element of D
for d9 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D st p = <*d1,d2*> holds
F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*>
let d1, d2 be Element of D; for d9 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D st p = <*d1,d2*> holds
F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*>
let d9 be Element of D9; for F being Function of [:D,D9:],E
for p being FinSequence of D st p = <*d1,d2*> holds
F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*>
let F be Function of [:D,D9:],E; for p being FinSequence of D st p = <*d1,d2*> holds
F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*>
let p be FinSequence of D; ( p = <*d1,d2*> implies F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*> )
assume A1:
p = <*d1,d2*>
; F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*>
A2:
p . 2 = d2
by A1, FINSEQ_1:44;
reconsider r = F [:] (p,d9) as FinSequence of E by Th81;
len p = 2
by A1, FINSEQ_1:44;
then A3:
len r = 2
by Th82;
then
2 in Seg (len r)
;
then
2 in dom r
by FINSEQ_1:def 3;
then A4:
r . 2 = F . (d2,d9)
by A2, FUNCOP_1:27;
1 in Seg (len r)
by A3;
then A5:
1 in dom r
by FINSEQ_1:def 3;
p . 1 = d1
by A1, FINSEQ_1:44;
then
r . 1 = F . (d1,d9)
by A5, FUNCOP_1:27;
hence
F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9))*>
by A3, A4, FINSEQ_1:44; verum