let E be set ; :: thesis: for A being Subset of (E ^omega )
for m, n being Nat holds (A * ) |^ m,n c= A *
let A be Subset of (E ^omega ); :: thesis: for m, n being Nat holds (A * ) |^ m,n c= A *
let m, n be Nat; :: thesis: (A * ) |^ m,n c= A *
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (A * ) |^ m,n or x in A * )
assume x in (A * ) |^ m,n ; :: thesis: x in A *
then consider mn being Nat such that
m <= mn and
mn <= n and
A1: x in (A * ) |^ mn by Th19;
(A * ) |^ mn c= A * by FLANG_1:66;
hence x in A * by A1; :: thesis: verum