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

theorem
  (A?) ^^ (A |^ k) = (A |^ k) ^^ (A?)
proof
  thus (A?) ^^ (A |^ k) = ({<%>E} \/ A) ^^ (A |^ k) by Th76
    .= ({<%>E} ^^ (A |^ k)) \/ (A ^^ (A |^ k)) by FLANG_1:20
    .= ({<%>E} ^^ (A |^ k)) \/ ((A |^ k) ^^ A) by FLANG_1:32
    .= (A |^ k) \/ ((A |^ k) ^^ A) by FLANG_1:13
    .= ((A |^ k) ^^ {<%>E}) \/ ((A |^ k) ^^ A) by FLANG_1:13
    .= (A |^ k) ^^ (A \/ {<%>E}) by FLANG_1:20
    .= (A |^ k) ^^ (A?) by Th76;
end;
