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

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

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

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

thus (A +) ^^ (A |^.. k) = A |^.. (k + 1) by Th90

.= (A |^.. k) ^^ (A |^.. 1) by Th18

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

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

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

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

thus (A +) ^^ (A |^.. k) = A |^.. (k + 1) by Th90

.= (A |^.. k) ^^ (A |^.. 1) by Th18

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