let E be set ; :: thesis: for A being Subset of (E ^omega)
for k, n being Nat holds A |^.. (n + k) = (A |^.. n) ^^ (A |^ k)
let A be Subset of (E ^omega); :: thesis: for k, n being Nat holds A |^.. (n + k) = (A |^.. n) ^^ (A |^ k)
let k, n be Nat; :: thesis: A |^.. (n + k) = (A |^.. n) ^^ (A |^ k)
defpred S₁[ Nat] means A |^.. (n + $1) = (A |^.. n) ^^ (A |^ $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 |^.. (n + (k + 1)) = A |^.. ((n + k) + 1)
.= ((A |^.. n) ^^ (A |^ k)) ^^ A by A2, Th16
.= (A |^.. n) ^^ ((A |^ k) ^^ A) by FLANG_1:18
.= (A |^.. n) ^^ (A |^ (k + 1)) by FLANG_1:23 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A |^.. (n + 0) = (A |^.. n) ^^ {(<%> E)} by FLANG_1:13
.= (A |^.. n) ^^ (A |^ 0) by FLANG_1:24 ;
then A3: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A3, A1);
hence A |^.. (n + k) = (A |^.. n) ^^ (A |^ k) ; :: thesis: verum