let k be Element of NAT ; :: thesis: for X being set
for A1, A2 being SetSequence of X holds (A1 (\) A2) ^\ k = (A1 ^\ k) (\) (A2 ^\ k)
let X be set ; :: thesis: for A1, A2 being SetSequence of X holds (A1 (\) A2) ^\ k = (A1 ^\ k) (\) (A2 ^\ k)
let A1, A2 be SetSequence of X; :: thesis: (A1 (\) A2) ^\ k = (A1 ^\ k) (\) (A2 ^\ k)

now

let n be Element of NAT ; :: thesis: ((A1 (\) A2) ^\ k) . n = ((A1 ^\ k) (\) (A2 ^\ k)) . n
thus ((A1 (\) A2) ^\ k) . n = (A1 (\) A2) . (n + k) by NAT_1:def 3
.= (A1 . (n + k)) \ (A2 . (n + k)) by Def3
.= ((A1 ^\ k) . n) \ (A2 . (n + k)) by NAT_1:def 3
.= ((A1 ^\ k) . n) \ ((A2 ^\ k) . n) by NAT_1:def 3
.= ((A1 ^\ k) (\) (A2 ^\ k)) . n by Def3 ; :: thesis: verum

end;

hence (A1 (\) A2) ^\ k = (A1 ^\ k) (\) (A2 ^\ k) by FUNCT_2:113; :: thesis: verum