let E be set ; :: thesis: for A being Subset of (E ^omega )
for k being Nat holds (A * ) ^^ (A |^ k) = A |^.. k
let A be Subset of (E ^omega ); :: thesis: for k being Nat holds (A * ) ^^ (A |^ k) = A |^.. k
let k be Nat; :: thesis: (A * ) ^^ (A |^ k) = A |^.. k
defpred S₁[ Nat] means (A * ) ^^ (A |^ $1) = A |^.. $1;

A1: now

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
(A * ) ^^ (A |^ (k + 1)) = (A * ) ^^ ((A |^ k) ^^ A) by FLANG_1:24
.= (A |^.. k) ^^ A by A2, FLANG_1:19
.= A |^.. (k + 1) by Th16 ;
hence S₁[k + 1] ; :: thesis: verum

end;

(A * ) ^^ (A |^ 0 ) = (A * ) ^^ {(<%> E)} by FLANG_1:25
.= A * by FLANG_1:14
.= A |^.. 0 by Th11 ;
then A3: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A1);
hence (A * ) ^^ (A |^ k) = A |^.. k ; :: thesis: verum