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 Th66:
  A |^ (m, n) ^^ (A*) = A* ^^ (A |^ (m, n))
proof
A1: A* ^^ (A |^ (m, n)) c= A |^ (m, n) ^^ (A*)
  proof
    let x be object;
    assume x in A* ^^ (A |^ (m, n));
    then consider a, b such that
A2: a in A* and
A3: b in A |^ (m, n) and
A4: x = a ^ b by FLANG_1:def 1;
    consider k such that
A5: a in A |^ k by A2,FLANG_1:41;
    consider mn such that
A6: m <= mn & mn <= n and
A7: b in A |^ mn by A3,Th19;
    A |^ k c= A* & A |^ mn c= A |^ (m, n) by A6,Th20,FLANG_1:42;
    then
A8: A |^ mn ^^ (A |^ k) c= A |^ (m, n) ^^ (A*) by FLANG_1:17;
    a ^ b in A |^ (mn + k) by A7,A5,FLANG_1:40;
    then a ^ b in A |^ mn ^^ (A |^ k) by FLANG_1:33;
    hence thesis by A4,A8;
  end;
  A |^ (m, n) ^^ (A*) c= A* ^^ (A |^ (m, n))
  proof
    let x be object;
    assume x in A |^ (m, n) ^^ (A*);
    then consider a, b such that
A9: a in A |^ (m, n) and
A10: b in A* and
A11: x = a ^ b by FLANG_1:def 1;
    consider k such that
A12: b in A |^ k by A10,FLANG_1:41;
    consider mn such that
A13: m <= mn & mn <= n and
A14: a in A |^ mn by A9,Th19;
    A |^ k c= A* & A |^ mn c= A |^ (m, n) by A13,Th20,FLANG_1:42;
    then
A15: A |^ k ^^ (A |^ mn) c= A* ^^ (A |^ (m, n)) by FLANG_1:17;
    a ^ b in A |^ (k + mn) by A14,A12,FLANG_1:40;
    then a ^ b in A |^ k ^^ (A |^ mn) by FLANG_1:33;
    hence thesis by A11,A15;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
