let k be Element of NAT ; :: thesis: for X being set
for A being Subset of X
for A1 being SetSequence of X holds (A (\) A1) ^\ k = A (\) (A1 ^\ k)
let X be set ; :: thesis: for A being Subset of X
for A1 being SetSequence of X holds (A (\) A1) ^\ k = A (\) (A1 ^\ k)
let A be Subset of X; :: thesis: for A1 being SetSequence of X holds (A (\) A1) ^\ k = A (\) (A1 ^\ k)
let A1 be SetSequence of X; :: thesis: (A (\) A1) ^\ k = A (\) (A1 ^\ k)

now

let n be Element of NAT ; :: thesis: ((A (\) A1) ^\ k) . n = (A (\) (A1 ^\ k)) . n
thus ((A (\) A1) ^\ k) . n = (A (\) A1) . (n + k) by NAT_1:def 3
.= A \ (A1 . (n + k)) by Def7
.= A \ ((A1 ^\ k) . n) by NAT_1:def 3
.= (A (\) (A1 ^\ k)) . n by Def7 ; :: thesis: verum

end;

hence (A (\) A1) ^\ k = A (\) (A1 ^\ k) by FUNCT_2:113; :: thesis: verum