let E be set ; :: thesis: for A being Subset of (E ^omega)
for k, m, n being Nat st k <= m holds
A |^ (m,n) c= A |^.. k
let A be Subset of (E ^omega); :: thesis: for k, m, n being Nat st k <= m holds
A |^ (m,n) c= A |^.. k
let k, m, n be Nat; :: thesis: ( k <= m implies A |^ (m,n) c= A |^.. k )
assume A1: k <= m ; :: thesis: A |^ (m,n) c= A |^.. k
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in A |^ (m,n) or x in A |^.. k )
assume x in A |^ (m,n) ; :: thesis: x in A |^.. k
then consider l being Nat such that
A2: m <= l and
l <= n and
A3: x in A |^ l by FLANG_2:19;
k <= l by A1, A2, XXREAL_0:2;
hence x in A |^.. k by A3, Th2; :: thesis: verum