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

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

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

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