let D, D', E be non empty set ; :: thesis: for d1, d2, d3 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
for q being FinSequence of D' st p = <*d1,d2,d3*> & q = <*d1',d2',d3'*> holds
F .: p,q = <*(F . d1,d1'),(F . d2,d2'),(F . d3,d3')*>
let d1, d2, d3 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
for q being FinSequence of D' st p = <*d1,d2,d3*> & q = <*d1',d2',d3'*> holds
F .: p,q = <*(F . d1,d1'),(F . d2,d2'),(F . d3,d3')*>
let d1', d2', d3' be Element of D'; :: thesis: for F being Function of [:D,D':],E
for p being FinSequence of D
for q being FinSequence of D' st p = <*d1,d2,d3*> & q = <*d1',d2',d3'*> holds
F .: p,q = <*(F . d1,d1'),(F . d2,d2'),(F . d3,d3')*>
let F be Function of [:D,D':],E; :: thesis: for p being FinSequence of D
for q being FinSequence of D' st p = <*d1,d2,d3*> & q = <*d1',d2',d3'*> holds
F .: p,q = <*(F . d1,d1'),(F . d2,d2'),(F . d3,d3')*>
let p be FinSequence of D; :: thesis: for q being FinSequence of D' st p = <*d1,d2,d3*> & q = <*d1',d2',d3'*> holds
F .: p,q = <*(F . d1,d1'),(F . d2,d2'),(F . d3,d3')*>
let q be FinSequence of D'; :: thesis: ( p = <*d1,d2,d3*> & q = <*d1',d2',d3'*> implies F .: p,q = <*(F . d1,d1'),(F . d2,d2'),(F . d3,d3')*> )
assume A1: ( p = <*d1,d2,d3*> & q = <*d1',d2',d3'*> ) ; :: thesis: F .: p,q = <*(F . d1,d1'),(F . d2,d2'),(F . d3,d3')*>
A2: ( p . 2 = d2 & q . 2 = d2' ) by A1, FINSEQ_1:62;
reconsider r = F .: p,q as FinSequence of E by Th84;
( len p = 3 & len q = 3 ) by A1, FINSEQ_1:62;
then A3: len r = 3 by Th86;
then 2 in Seg (len r) ;
then 2 in dom r by FINSEQ_1:def 3;
then A4: r . 2 = F . d2,d2' by A2, FUNCOP_1:28;
A5: ( p . 3 = d3 & q . 3 = d3' ) by A1, FINSEQ_1:62;
A6: ( p . 1 = d1 & q . 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 . d3,d3' by A5, FUNCOP_1:28;
1 in Seg (len r) by A3;
then 1 in dom r by FINSEQ_1:def 3;
then r . 1 = F . d1,d1' by A6, FUNCOP_1:28;
hence F .: p,q = <*(F . d1,d1'),(F . d2,d2'),(F . d3,d3')*> by A3, A4, A7, FINSEQ_1:62; :: thesis: verum