let D, D9, E be non empty set ; :: thesis: for d1 being Element of D
for d19 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*> & q = <*d19*> holds
F .: (p,q) = <*(F . (d1,d19))*>
let d1 be Element of D; :: thesis: for d19 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*> & q = <*d19*> holds
F .: (p,q) = <*(F . (d1,d19))*>
let d19 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*> & q = <*d19*> holds
F .: (p,q) = <*(F . (d1,d19))*>
let F be Function of [:D,D9:],E; :: thesis: for p being FinSequence of D
for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds
F .: (p,q) = <*(F . (d1,d19))*>
let p be FinSequence of D; :: thesis: for q being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds
F .: (p,q) = <*(F . (d1,d19))*>
let q be FinSequence of D9; :: thesis: ( p = <*d1*> & q = <*d19*> implies F .: (p,q) = <*(F . (d1,d19))*> )
assume A1: ( p = <*d1*> & q = <*d19*> ) ; :: thesis: F .: (p,q) = <*(F . (d1,d19))*>
A2: ( p . 1 = d1 & q . 1 = d19 ) by A1;
reconsider r = F .: (p,q) as FinSequence of E by Th68;
( len p = 1 & len q = 1 ) by A1, FINSEQ_1:39;
then A3: len r = 1 by Th70;
then 1 in Seg (len r) ;
then 1 in dom r by FINSEQ_1:def 3;
then r . 1 = F . (d1,d19) by A2, FUNCOP_1:22;
hence F .: (p,q) = <*(F . (d1,d19))*> by A3, FINSEQ_1:40; :: thesis: verum