let D, D9, E be non empty set ; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( 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*> ) ; :: thesis: F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29)),(F . (d3,d39))*>
A2: ( p . 2 = d2 & q . 2 = d29 ) by A1;
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;
A6: ( p . 1 = d1 & q . 1 = d19 ) by A1;
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; :: thesis: verum