let D, D', E be non empty set ; :: thesis: for d being Element of D
for d1', d2', d3' being Element of D'
for F being Function of [:D,D':],E
for p being FinSequence of D' st p = <*d1',d2',d3'*> holds
F [;] d,p = <*(F . d,d1'),(F . d,d2'),(F . d,d3')*>
let d be Element of D; :: thesis: for d1', d2', d3' being Element of D'
for F being Function of [:D,D':],E
for p being FinSequence of D' st p = <*d1',d2',d3'*> holds
F [;] d,p = <*(F . d,d1'),(F . d,d2'),(F . d,d3')*>
let d1', d2', d3' be Element of D'; :: thesis: for F being Function of [:D,D':],E
for p being FinSequence of D' st p = <*d1',d2',d3'*> holds
F [;] d,p = <*(F . d,d1'),(F . d,d2'),(F . d,d3')*>
let F be Function of [:D,D':],E; :: thesis: for p being FinSequence of D' st p = <*d1',d2',d3'*> holds
F [;] d,p = <*(F . d,d1'),(F . d,d2'),(F . d,d3')*>
let p be FinSequence of D'; :: thesis: ( p = <*d1',d2',d3'*> implies F [;] d,p = <*(F . d,d1'),(F . d,d2'),(F . d,d3')*> )
assume A1: p = <*d1',d2',d3'*> ; :: thesis: F [;] d,p = <*(F . d,d1'),(F . d,d2'),(F . d,d3')*>
A2: p . 2 = d2' by A1, FINSEQ_1:62;
reconsider r = F [;] d,p as FinSequence of E by Th91;
len p = 3 by A1, FINSEQ_1:62;
then A3: len r = 3 by Th92;
then 2 in Seg (len r) ;
then 2 in dom r by FINSEQ_1:def 3;
then A4: r . 2 = F . d,d2' by A2, FUNCOP_1:42;
A5: p . 3 = d3' by A1, FINSEQ_1:62;
A6: p . 1 = d1' by A1, FINSEQ_1:62;
3 in Seg (len r) by A3;
then 3 in dom r by FINSEQ_1:def 3;
then A7: r . 3 = F . d,d3' by A5, FUNCOP_1:42;
1 in Seg (len r) by A3;
then 1 in dom r by FINSEQ_1:def 3;
then r . 1 = F . d,d1' by A6, FUNCOP_1:42;
hence F [;] d,p = <*(F . d,d1'),(F . d,d2'),(F . d,d3')*> by A3, A4, A7, FINSEQ_1:62; :: thesis: verum