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 *

let A, B be Subset of (E ^omega); :: thesis: for n being Nat st A c= B * holds

A |^.. n c= B *

let n be Nat; :: thesis: ( A c= B * implies A |^.. n c= B * )

assume A1: A c= B * ; :: thesis: A |^.. n c= B *

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in A |^.. n or x in B * )

assume x in A |^.. n ; :: thesis: x in B *

then consider k being Nat such that

k >= n and

A2: x in A |^ k by Th2;

A |^ k c= B * by A1, FLANG_1:59;

hence x in B * by A2; :: thesis: verum

for n being Nat st A c= B * holds

A |^.. n c= B *

let A, B be Subset of (E ^omega); :: thesis: for n being Nat st A c= B * holds

A |^.. n c= B *

let n be Nat; :: thesis: ( A c= B * implies A |^.. n c= B * )

assume A1: A c= B * ; :: thesis: A |^.. n c= B *

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in A |^.. n or x in B * )

assume x in A |^.. n ; :: thesis: x in B *

then consider k being Nat such that

k >= n and

A2: x in A |^ k by Th2;

A |^ k c= B * by A1, FLANG_1:59;

hence x in B * by A2; :: thesis: verum