let E be set ; :: thesis: for A being Subset of (E ^omega )
for k being Nat holds (A ? ) ^^ (A |^ k) = A |^ k,(k + 1)
let A be Subset of (E ^omega ); :: thesis: for k being Nat holds (A ? ) ^^ (A |^ k) = A |^ k,(k + 1)
let k be Nat; :: thesis: (A ? ) ^^ (A |^ k) = A |^ k,(k + 1)
(A |^ 0 ,1) ^^ (A |^ k) = (A |^ 0 ,1) ^^ (A |^ k,k) by Th22
.= A |^ (0 + k),(1 + k) by Th37 ;
hence (A ? ) ^^ (A |^ k) = A |^ k,(k + 1) by Th79; :: thesis: verum