let E be set ; :: thesis: for A being Subset of (E ^omega)

for k being Nat holds (A |^ k) ^^ (A +) = (A +) ^^ (A |^ k)

let A be Subset of (E ^omega); :: thesis: for k being Nat holds (A |^ k) ^^ (A +) = (A +) ^^ (A |^ k)

let k be Nat; :: thesis: (A |^ k) ^^ (A +) = (A +) ^^ (A |^ k)

thus (A |^ k) ^^ (A +) = (A |^ k) ^^ (A |^.. 1) by Th50

.= (A |^.. 1) ^^ (A |^ k) by Th24

.= (A +) ^^ (A |^ k) by Th50 ; :: thesis: verum

for k being Nat holds (A |^ k) ^^ (A +) = (A +) ^^ (A |^ k)

let A be Subset of (E ^omega); :: thesis: for k being Nat holds (A |^ k) ^^ (A +) = (A +) ^^ (A |^ k)

let k be Nat; :: thesis: (A |^ k) ^^ (A +) = (A +) ^^ (A |^ k)

thus (A |^ k) ^^ (A +) = (A |^ k) ^^ (A |^.. 1) by Th50

.= (A |^.. 1) ^^ (A |^ k) by Th24

.= (A +) ^^ (A |^ k) by Th50 ; :: thesis: verum