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')*>
reconsider r = F [;] d,p as FinSequence of E by Th91;
len p = 3 by A1, FINSEQ_1:62;
then A2: len r = 3 by Th92;
then ( 1 in Seg (len r) & 2 in Seg (len r) & 3 in Seg (len r) ) ;
then ( 1 in dom r & 2 in dom r & 3 in dom r & p . 1 = d1' & p . 2 = d2' & p . 3 = d3' ) by A1, FINSEQ_1:62, FINSEQ_1:def 3;
then ( r . 1 = F . d,d1' & r . 2 = F . d,d2' & r . 3 = F . d,d3' ) by FUNCOP_1:42;
hence F [;] d,p = <*(F . d,d1'),(F . d,d2'),(F . d,d3')*> by A2, FINSEQ_1:62; :: thesis: verum