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 Th17:
  (A |^.. m) ^^ (A*) = A |^.. m
proof
  now
    let x be object;
    assume x in (A |^.. m) ^^ (A*);
    then consider a, b such that
A1: a in A |^.. m and
A2: b in A* and
A3: x = a ^ b by FLANG_1:def 1;
    consider k such that
A4: b in A |^ k by A2,FLANG_1:41;
    consider l such that
A5: m <= l and
A6: a in A |^ l by A1,Th2;
A7: l + k >= m by A5,NAT_1:12;
    a ^ b in A |^ (l + k) by A4,A6,FLANG_1:40;
    hence x in A |^.. m by A3,A7,Th2;
  end;
  then
A8: (A |^.. m) ^^ (A*) c= A |^.. m;
  <%>E in A* by FLANG_1:48;
  then A |^.. m c= (A |^.. m) ^^ (A*) by FLANG_1:16;
  hence thesis by A8,XBOOLE_0:def 10;
end;
