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

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 Th16
.= ((A |^.. k) ^^ A) ^^ A by A2, FLANG_1:19
.= (A |^.. (k + 1)) ^^ A by Th16 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A ^^ (A |^.. 0 ) = A ^^ (A * ) by Th11
.= (A * ) ^^ A by FLANG_1:58
.= (A |^.. 0 ) ^^ A by Th11 ;
then A3: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A1);
hence A ^^ (A |^.. n) = (A |^.. n) ^^ A ; :: thesis: verum