let D, D9, E be non empty set ; :: thesis: for d1, d2 being Element of D
for d19, d29 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*> & q = <*d19,d29*> holds
F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*>
let d1, d2 be Element of D; :: thesis: for d19, d29 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*> & q = <*d19,d29*> holds
F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*>
let d19, d29 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*> & q = <*d19,d29*> holds
F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*>
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*> & q = <*d19,d29*> holds
F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*>
let p be FinSequence of D; :: thesis: for q being FinSequence of D9 st p = <*d1,d2*> & q = <*d19,d29*> holds
F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*>
let q be FinSequence of D9; :: thesis: ( p = <*d1,d2*> & q = <*d19,d29*> implies F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*> )
assume A1: ( p = <*d1,d2*> & q = <*d19,d29*> ) ; :: thesis: F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*>
A2: ( p . 2 = d2 & q . 2 = d29 ) by A1;
reconsider r = F .: (p,q) as FinSequence of E by Th68;
( len p = 2 & len q = 2 ) by A1, FINSEQ_1:44;
then A3: len r = 2 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;
1 in Seg (len r) by A3;
then A5: 1 in dom r by FINSEQ_1:def 3;
( p . 1 = d1 & q . 1 = d19 ) by A1;
then r . 1 = F . (d1,d19) by A5, FUNCOP_1:22;
hence F .: (p,q) = <*(F . (d1,d19)),(F . (d2,d29))*> by A3, A4, FINSEQ_1:44; :: thesis: verum