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

now

let x be set ; :: thesis: ( x in A |^.. n implies x in A |^.. m )
assume x in A |^.. n ; :: thesis: x in A |^.. m
then consider k being Nat such that
A2: n <= k and
A3: x in A |^ k by Th2;
m <= k by A1, A2, XXREAL_0:2;
hence x in A |^.. m by A3, Th2; :: thesis: verum

end;

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