let D, D', E be non empty set ; :: thesis: for d being Element of D
for d1', d2' being Element of D'
for F being Function of [:D,D':],E
for p being FinSequence of D' st p = <*d1',d2'*> holds
F [;] d,p = <*(F . d,d1'),(F . d,d2')*>
let d be Element of D; :: thesis: for d1', d2' being Element of D'
for F being Function of [:D,D':],E
for p being FinSequence of D' st p = <*d1',d2'*> holds
F [;] d,p = <*(F . d,d1'),(F . d,d2')*>
let d1', d2' be Element of D'; :: thesis: for F being Function of [:D,D':],E
for p being FinSequence of D' st p = <*d1',d2'*> holds
F [;] d,p = <*(F . d,d1'),(F . d,d2')*>
let F be Function of [:D,D':],E; :: thesis: for p being FinSequence of D' st p = <*d1',d2'*> holds
F [;] d,p = <*(F . d,d1'),(F . d,d2')*>
let p be FinSequence of D'; :: thesis: ( p = <*d1',d2'*> implies F [;] d,p = <*(F . d,d1'),(F . d,d2')*> )
assume A1: p = <*d1',d2'*> ; :: thesis: F [;] d,p = <*(F . d,d1'),(F . d,d2')*>
A2: p . 2 = d2' by A1, FINSEQ_1:61;
reconsider r = F [;] d,p as FinSequence of E by Th91;
len p = 2 by A1, FINSEQ_1:61;
then A3: len r = 2 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;
1 in Seg (len r) by A3;
then A5: 1 in dom r by FINSEQ_1:def 3;
p . 1 = d1' by A1, FINSEQ_1:61;
then r . 1 = F . d,d1' by A5, FUNCOP_1:42;
hence F [;] d,p = <*(F . d,d1'),(F . d,d2')*> by A3, A4, FINSEQ_1:61; :: thesis: verum