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

for d9 being Element of D9

for i being natural Number

for F being Function of [:D,D9:],E holds F [:] ((i |-> d),d9) = i |-> (F . (d,d9))

let d be Element of D; :: thesis: for d9 being Element of D9

for i being natural Number

for F being Function of [:D,D9:],E holds F [:] ((i |-> d),d9) = i |-> (F . (d,d9))

let d9 be Element of D9; :: thesis: for i being natural Number

for F being Function of [:D,D9:],E holds F [:] ((i |-> d),d9) = i |-> (F . (d,d9))

let i be natural Number ; :: thesis: for F being Function of [:D,D9:],E holds F [:] ((i |-> d),d9) = i |-> (F . (d,d9))

let F be Function of [:D,D9:],E; :: thesis: F [:] ((i |-> d),d9) = i |-> (F . (d,d9))

thus F [:] ((i |-> d),d9) = F .: ((i |-> d),(i |-> d9))

.= i |-> (F . (d,d9)) by Th17 ; :: thesis: verum

for d9 being Element of D9

for i being natural Number

for F being Function of [:D,D9:],E holds F [:] ((i |-> d),d9) = i |-> (F . (d,d9))

let d be Element of D; :: thesis: for d9 being Element of D9

for i being natural Number

for F being Function of [:D,D9:],E holds F [:] ((i |-> d),d9) = i |-> (F . (d,d9))

let d9 be Element of D9; :: thesis: for i being natural Number

for F being Function of [:D,D9:],E holds F [:] ((i |-> d),d9) = i |-> (F . (d,d9))

let i be natural Number ; :: thesis: for F being Function of [:D,D9:],E holds F [:] ((i |-> d),d9) = i |-> (F . (d,d9))

let F be Function of [:D,D9:],E; :: thesis: F [:] ((i |-> d),d9) = i |-> (F . (d,d9))

thus F [:] ((i |-> d),d9) = F .: ((i |-> d),(i |-> d9))

.= i |-> (F . (d,d9)) by Th17 ; :: thesis: verum