let D, D9, E be non empty set ; :: thesis: for d being Element of D

for i being natural Number

for F being Function of [:D,D9:],E

for T being Tuple of i,D holds F [:] (T,d) = (F [:] ((id D),d)) * T

let d be Element of D; :: thesis: for i being natural Number

for F being Function of [:D,D9:],E

for T being Tuple of i,D holds F [:] (T,d) = (F [:] ((id D),d)) * T

let i be natural Number ; :: thesis: for F being Function of [:D,D9:],E

for T being Tuple of i,D holds F [:] (T,d) = (F [:] ((id D),d)) * T

let F be Function of [:D,D9:],E; :: thesis: for T being Tuple of i,D holds F [:] (T,d) = (F [:] ((id D),d)) * T

let T be Tuple of i,D; :: thesis: F [:] (T,d) = (F [:] ((id D),d)) * T

rng T c= D ;

hence F [:] (T,d) = F [:] (((id D) * T),d) by RELAT_1:53

.= (F [:] ((id D),d)) * T by FUNCOP_1:29 ;

:: thesis: verum

for i being natural Number

for F being Function of [:D,D9:],E

for T being Tuple of i,D holds F [:] (T,d) = (F [:] ((id D),d)) * T

let d be Element of D; :: thesis: for i being natural Number

for F being Function of [:D,D9:],E

for T being Tuple of i,D holds F [:] (T,d) = (F [:] ((id D),d)) * T

let i be natural Number ; :: thesis: for F being Function of [:D,D9:],E

for T being Tuple of i,D holds F [:] (T,d) = (F [:] ((id D),d)) * T

let F be Function of [:D,D9:],E; :: thesis: for T being Tuple of i,D holds F [:] (T,d) = (F [:] ((id D),d)) * T

let T be Tuple of i,D; :: thesis: F [:] (T,d) = (F [:] ((id D),d)) * T

rng T c= D ;

hence F [:] (T,d) = F [:] (((id D) * T),d) by RELAT_1:53

.= (F [:] ((id D),d)) * T by FUNCOP_1:29 ;

:: thesis: verum