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 T9 being Tuple of i,D9 holds F .: ((i |-> d),T9) = F [;] (d,T9)

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

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

for T9 being Tuple of i,D9 holds F .: ((i |-> d),T9) = F [;] (d,T9)

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

for T9 being Tuple of i,D9 holds F .: ((i |-> d),T9) = F [;] (d,T9)

let F be Function of [:D,D9:],E; :: thesis: for T9 being Tuple of i,D9 holds F .: ((i |-> d),T9) = F [;] (d,T9)

let T9 be Tuple of i,D9; :: thesis: F .: ((i |-> d),T9) = F [;] (d,T9)

dom T9 = Seg (len T9) by FINSEQ_1:def 3

.= Seg i by CARD_1:def 7 ;

hence F .: ((i |-> d),T9) = F [;] (d,T9) ; :: thesis: verum

for i being natural Number

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

for T9 being Tuple of i,D9 holds F .: ((i |-> d),T9) = F [;] (d,T9)

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

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

for T9 being Tuple of i,D9 holds F .: ((i |-> d),T9) = F [;] (d,T9)

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

for T9 being Tuple of i,D9 holds F .: ((i |-> d),T9) = F [;] (d,T9)

let F be Function of [:D,D9:],E; :: thesis: for T9 being Tuple of i,D9 holds F .: ((i |-> d),T9) = F [;] (d,T9)

let T9 be Tuple of i,D9; :: thesis: F .: ((i |-> d),T9) = F [;] (d,T9)

dom T9 = Seg (len T9) by FINSEQ_1:def 3

.= Seg i by CARD_1:def 7 ;

hence F .: ((i |-> d),T9) = F [;] (d,T9) ; :: thesis: verum