let D, D9, E be non empty set ; :: thesis: for d1, d2, d3 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,d3*> holds
F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*>
let d1, d2, d3 be Element of D; :: thesis: for d9 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D st p = <*d1,d2,d3*> holds
F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*>
let d9 be Element of D9; :: thesis: for F being Function of [:D,D9:],E
for p being FinSequence of D st p = <*d1,d2,d3*> holds
F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*>
let F be Function of [:D,D9:],E; :: thesis: for p being FinSequence of D st p = <*d1,d2,d3*> holds
F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*>
let p be FinSequence of D; :: thesis: ( p = <*d1,d2,d3*> implies F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*> )
assume A1: p = <*d1,d2,d3*> ; :: thesis: F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*>
A2: p . 2 = d2 by A1;
reconsider r = F [:] (p,d9) as FinSequence of E by Th81;
len p = 3 by A1, FINSEQ_1:45;
then A3: len r = 3 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;
A5: p . 3 = d3 by A1;
A6: p . 1 = d1 by A1;
3 in Seg (len r) by A3;
then 3 in dom r by FINSEQ_1:def 3;
then A7: r . 3 = F . (d3,d9) by A5, FUNCOP_1:27;
1 in Seg (len r) by A3;
then 1 in dom r by FINSEQ_1:def 3;
then r . 1 = F . (d1,d9) by A6, FUNCOP_1:27;
hence F [:] (p,d9) = <*(F . (d1,d9)),(F . (d2,d9)),(F . (d3,d9))*> by A3, A4, A7, FINSEQ_1:45; :: thesis: verum