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 Th35:
  A |^ (m, n) ^^ (A |^ k) = (A |^ k) ^^ (A |^ (m, n))
proof
A1: now
    let x be object;
    assume x in (A |^ k) ^^ (A |^ (m, n));
    then consider a, b such that
A2: a in (A |^ k) and
A3: b in A |^ (m, n) and
A4: x = a ^ b by FLANG_1:def 1;
    consider mn such that
A5: m <= mn & mn <= n and
A6: b in A |^ mn by A3,Th19;
    A |^ mn c= A |^ (m, n) by A5,Th20;
    then
A7: (A |^ mn) ^^ (A |^ k) c= (A |^ (m, n)) ^^ (A |^ k) by FLANG_1:17;
    a ^ b in (A |^ k) ^^ (A |^ mn) by A2,A6,FLANG_1:def 1;
    then a ^ b in (A |^ mn) ^^ (A |^ k) by Th12;
    hence x in (A |^ (m, n)) ^^ (A |^ k) by A4,A7;
  end;
  now
    let x be object;
    assume x in (A |^ (m, n)) ^^ (A |^ k);
    then consider a, b such that
A8: a in A |^ (m, n) and
A9: b in (A |^ k) and
A10: x = a ^ b by FLANG_1:def 1;
    consider mn such that
A11: m <= mn & mn <= n and
A12: a in A |^ mn by A8,Th19;
    A |^ mn c= A |^ (m, n) by A11,Th20;
    then
A13: (A |^ k) ^^ (A |^ mn) c= (A |^ k) ^^ (A |^ (m, n)) by FLANG_1:17;
    a ^ b in (A |^ mn) ^^ (A |^ k) by A9,A12,FLANG_1:def 1;
    then a ^ b in (A |^ k) ^^ (A |^ mn) by Th12;
    hence x in (A |^ k) ^^ (A |^ (m, n)) by A10,A13;
  end;
  hence thesis by A1,TARSKI:2;
end;
