let E be set ; :: thesis: for A being Subset of (E ^omega)
for k being Nat holds (A ?) |^ k = A |^ (0,k)
let A be Subset of (E ^omega); :: thesis: for k being Nat holds (A ?) |^ k = A |^ (0,k)
let k be Nat; :: thesis: (A ?) |^ k = A |^ (0,k)
defpred S₁[ Nat] means (A ?) |^ $1 = A |^ (0,$1);

A1: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
(A ?) |^ (k + 1) = ((A ?) |^ k) ^^ ((A ?) |^ 1) by FLANG_1:33
.= (A |^ (0,k)) ^^ (A ?) by A2, FLANG_1:25
.= (A |^ (0,k)) ^^ (A |^ (0,1)) by Th79
.= A |^ ((0 + 0),(k + 1)) by Th37 ;
hence S₁[k + 1] ; :: thesis: verum

end;

(A ?) |^ 0 = {(<%> E)} by FLANG_1:24
.= A |^ (0,0) by Th45 ;
then A3: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A1);
hence (A ?) |^ k = A |^ (0,k) ; :: thesis: verum