let k be 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)
let n be Element of NAT ; :: according to FUNCT_2:def 8 :: 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 Def2
.= ((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 Def2 ; :: thesis: verum