let E be set ; :: thesis: for A, B being Subset of (E ^omega)
for n being Nat st A c= B holds
A |^.. n c= B |^.. n
let A, B be Subset of (E ^omega); :: thesis: for n being Nat st A c= B holds
A |^.. n c= B |^.. n
let n be Nat; :: thesis: ( A c= B implies A |^.. n c= B |^.. n )
assume A1: A c= B ; :: thesis: A |^.. n c= B |^.. n

now

let x be set ; :: thesis: ( x in A |^.. n implies x in B |^.. n )
assume x in A |^.. n ; :: thesis: x in B |^.. n
then consider k being Nat such that
A2: n <= k and
A3: x in A |^ k by Th2;
A |^ k c= B |^ k by A1, FLANG_1:37;
hence x in B |^.. n by A2, A3, Th2; :: thesis: verum

end;

hence A |^.. n c= B |^.. n by TARSKI:def 3; :: thesis: verum