let x1, x2 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,x2*> holds
f * p = <*(f . x1),(f . x2)*>
let D, D9 be non empty set ; :: thesis: for p being FinSequence of D
for f being Function of D,D9 st p = <*x1,x2*> holds
f * p = <*(f . x1),(f . x2)*>
let p be FinSequence of D; :: thesis: for f being Function of D,D9 st p = <*x1,x2*> holds
f * p = <*(f . x1),(f . x2)*>
let f be Function of D,D9; :: thesis: ( p = <*x1,x2*> implies f * p = <*(f . x1),(f . x2)*> )
assume A1: p = <*x1,x2*> ; :: thesis: f * p = <*(f . x1),(f . x2)*>
A2: p . 2 = x2 by A1;
reconsider q = f * p as FinSequence of D9 by Th30;
len p = 2 by A1, FINSEQ_1:44;
then A3: len q = 2 by Th31;
then 2 in Seg (len q) ;
then 2 in dom q by FINSEQ_1:def 3;
then A4: q . 2 = f . x2 by A2, FUNCT_1:12;
1 in Seg (len q) by A3;
then A5: 1 in dom q by FINSEQ_1:def 3;
p . 1 = x1 by A1;
then q . 1 = f . x1 by A5, FUNCT_1:12;
hence f * p = <*(f . x1),(f . x2)*> by A3, A4, FINSEQ_1:44; :: thesis: verum