reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, a1, a2, b for Element of E^omega;
reserve i, k, l, m, n for Nat;

theorem Th7:
  m <= n + 1 implies A |^ (m, n) \/ A |^.. (n + 1) = A |^.. m
proof
  assume m <= n + 1;
  then
A1: A |^.. (n + 1) c= A |^.. m by Th5;
  now
    let x be object;
    assume x in A |^.. m;
    then consider k such that
A2: m <= k and
A3: x in A |^ k by Th2;
A4: now
      assume k > n;
      then k >= n + 1 by NAT_1:13;
      hence x in A |^.. (n + 1) by A3,Th2;
    end;
    k <= n implies x in A |^ (m, n) by A2,A3,FLANG_2:19;
    hence x in A |^ (m, n) \/ A |^.. (n + 1) by A4,XBOOLE_0:def 3;
  end;
  then
A5: A |^.. m c= A |^ (m, n) \/ A |^.. (n + 1);
  A |^ (m, n) c= A |^.. m by Th6;
  then A |^ (m, n) \/ A |^.. (n + 1) c= A |^.. m by A1,XBOOLE_1:8;
  hence thesis by A5,XBOOLE_0:def 10;
end;
