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 Th37:
  m <= n & k <= l implies (A |^ (m, n)) ^^ (A |^ (k, l)) = A |^ (m + k, n + l)
proof
  assume
A1: m <= n & k <= l;
A2: now
    let x be object;
    assume x in A |^ (m + k, n + l);
    then consider i such that
A3: m + k <= i & i <= n + l and
A4: x in A |^ i by Th19;
    consider mn, kl such that
A5: mn + kl = i and
A6: m <= mn & mn <= n & k <= kl & kl <= l by A1,A3,Th2;
    A |^ mn c= A |^ (m, n) & A |^ kl c= A |^ (k, l) by A6,Th20;
    then (A |^ mn) ^^ (A |^ kl) c= (A |^ (m, n)) ^^ (A |^ (k, l)) by FLANG_1:17
;
    then (A |^ (mn + kl)) c= (A |^ (m, n)) ^^ (A |^ (k, l)) by FLANG_1:33;
    hence x in (A |^ (m, n)) ^^ (A |^ (k, l)) by A4,A5;
  end;
  now
    let x be object;
    assume x in (A |^ (m, n)) ^^ (A |^ (k, l));
    then consider a, b such that
A7: a in A |^ (m, n) and
A8: b in A |^ (k, l) and
A9: x = a ^ b by FLANG_1:def 1;
    consider kl such that
A10: k <= kl and
A11: kl <= l and
A12: b in A |^ kl by A8,Th19;
    consider mn such that
A13: m <= mn and
A14: mn <= n and
A15: a in A |^ mn by A7,Th19;
A16: mn + kl <= n + l by A14,A11,XREAL_1:7;
    a ^ b in A |^ (mn + kl) & m + k <= mn + kl by A13,A15,A10,A12,FLANG_1:40
,XREAL_1:7;
    hence x in A |^ (m + k, n + l) by A9,A16,Th19;
  end;
  hence thesis by A2,TARSKI:2;
end;
