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

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in A |^ m,n or x in B |^ m,n )
assume x in A |^ m,n ; :: thesis: x in B |^ m,n
then consider k being Nat such that
A2: ( m <= k & k <= n & x in A |^ k ) by Th19;
A |^ k c= B |^ k by A1, FLANG_1:38;
hence x in B |^ m,n by A2, Th19; :: thesis: verum

end;