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')*>
reconsider r = F .: p,q as FinSequence of E by Th84;
( len p = 3 & len q = 3 ) by A1, FINSEQ_1:62;
then A2: len r = 3 by Th86;
then ( 1 in Seg (len r) & 2 in Seg (len r) & 3 in Seg (len r) ) ;
then ( 1 in dom r & 2 in dom r & 3 in dom r & p . 1 = d1 & q . 1 = d1' & p . 2 = d2 & q . 2 = d2' & p . 3 = d3 & q . 3 = d3' ) by A1, FINSEQ_1:62, FINSEQ_1:def 3;
then ( r . 1 = F . d1,d1' & r . 2 = F . d2,d2' & r . 3 = F . d3,d3' ) by FUNCOP_1:28;
hence F .: p,q = <*(F . d1,d1'),(F . d2,d2'),(F . d3,d3')*> by A2, FINSEQ_1:62; :: thesis: verum