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