let i be natural Number ; for D, D9, E being non empty set
for d9 being Element of D9
for F being Function of [:D,D9:],E
for z being Tuple of i,D holds F [:] (z,d9) is Element of i -tuples_on E
let D, D9, E be non empty set ; for d9 being Element of D9
for F being Function of [:D,D9:],E
for z being Tuple of i,D holds F [:] (z,d9) is Element of i -tuples_on E
let d9 be Element of D9; for F being Function of [:D,D9:],E
for z being Tuple of i,D holds F [:] (z,d9) is Element of i -tuples_on E
let F be Function of [:D,D9:],E; for z being Tuple of i,D holds F [:] (z,d9) is Element of i -tuples_on E
let z be Tuple of i,D; F [:] (z,d9) is Element of i -tuples_on E
reconsider r = F [:] (z,d9) as FinSequence of E by Th81;
len z = i
by CARD_1:def 7;
then
len r = i
by Th82;
hence
F [:] (z,d9) is Element of i -tuples_on E
by Th90; verum