let D, D9, E be non empty set ; for d1, d2, d3 being Element of D
for d19, d29, d39 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D
for q being FinSequence of D9 st p = <*d1,d2,d3*> & q = <*d19,d29,d39*> holds
F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*>
let d1, d2, d3 be Element of D; for d19, d29, d39 being Element of D9
for F being Function of [:D,D9:],E
for p being FinSequence of D
for q being FinSequence of D9 st p = <*d1,d2,d3*> & q = <*d19,d29,d39*> holds
F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*>
let d19, d29, d39 be Element of D9; for F being Function of [:D,D9:],E
for p being FinSequence of D
for q being FinSequence of D9 st p = <*d1,d2,d3*> & q = <*d19,d29,d39*> holds
F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*>
let F be Function of [:D,D9:],E; for p being FinSequence of D
for q being FinSequence of D9 st p = <*d1,d2,d3*> & q = <*d19,d29,d39*> holds
F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*>
let p be FinSequence of D; for q being FinSequence of D9 st p = <*d1,d2,d3*> & q = <*d19,d29,d39*> holds
F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*>
let q be FinSequence of D9; ( p = <*d1,d2,d3*> & q = <*d19,d29,d39*> implies F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*> )
assume A1:
( p = <*d1,d2,d3*> & q = <*d19,d29,d39*> )
; F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*>
A2:
( p . 2 = d2 & q . 2 = d29 )
by A1, FINSEQ_1:45;
reconsider r = F .: (p,q) as FinSequence of E by Th68;
( len p = 3 & len q = 3 )
by A1, FINSEQ_1:45;
then A3:
len r = 3
by Th70;
then
2 in Seg (len r)
;
then
2 in dom r
by FINSEQ_1:def 3;
then A4:
r . 2 = F . (d2,d29)
by A2, FUNCOP_1:22;
A5:
( p . 3 = d3 & q . 3 = d39 )
by A1, FINSEQ_1:45;
A6:
( p . 1 = d1 & q . 1 = d19 )
by A1, FINSEQ_1:45;
3 in Seg (len r)
by A3;
then
3 in dom r
by FINSEQ_1:def 3;
then A7:
r . 3 = F . (d3,d39)
by A5, FUNCOP_1:22;
1 in Seg (len r)
by A3;
then
1 in dom r
by FINSEQ_1:def 3;
then
r . 1 = F . (d1,d19)
by A6, FUNCOP_1:22;
hence
F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*>
by A3, A4, A7, FINSEQ_1:45; verum