let E be set ; :: thesis: for A being Subset of (E ^omega)
for m, n being Nat holds (A |^.. m) ^^ (A |^.. n) = (A |^.. n) ^^ (A |^.. m)
let A be Subset of (E ^omega); :: thesis: for m, n being Nat holds (A |^.. m) ^^ (A |^.. n) = (A |^.. n) ^^ (A |^.. m)
let m, n be Nat; :: thesis: (A |^.. m) ^^ (A |^.. n) = (A |^.. n) ^^ (A |^.. m)
thus (A |^.. m) ^^ (A |^.. n) = A |^.. (m + n) by Th18
.= (A |^.. n) ^^ (A |^.. m) by Th18 ; :: thesis: verum