let x1 be object ; :: thesis: for D, D9 being non empty set
for p being FinSequence of D
for f being Function of D,D9 st p = <*x1*> holds
f * p = <*(f . x1)*>
let D, D9 be non empty set ; :: thesis: for p being FinSequence of D
for f being Function of D,D9 st p = <*x1*> holds
f * p = <*(f . x1)*>
let p be FinSequence of D; :: thesis: for f being Function of D,D9 st p = <*x1*> holds
f * p = <*(f . x1)*>
let f be Function of D,D9; :: thesis: ( p = <*x1*> implies f * p = <*(f . x1)*> )
assume A1: p = <*x1*> ; :: thesis: f * p = <*(f . x1)*>
A2: p . 1 = x1 by A1, FINSEQ_1:40;
reconsider q = f * p as FinSequence of D9 by Th30;
len p = 1 by A1, FINSEQ_1:39;
then A3: len q = 1 by Th31;
then 1 in Seg (len q) ;
then 1 in dom q by FINSEQ_1:def 3;
then q . 1 = f . x1 by A2, FUNCT_1:12;
hence f * p = <*(f . x1)*> by A3, FINSEQ_1:40; :: thesis: verum