reserve E, x, y, X for set;
reserve A, B, C, D for Subset of E^omega;
reserve a, a1, a2, b, c, c1, c2, d, ab, bc for Element of E^omega;
reserve e for Element of E;
reserve i, j, k, l, n, n1, n2, m for Nat;

theorem Th33:
  A |^ (m + n) = (A |^ m) ^^ (A |^ n)
proof
  defpred P[Nat] means for m, n st m + n <= $1 holds A |^ (m + n) = (A |^ m)
  ^^ (A |^ n);
A1: now
    let l;
    assume
A2: P[l];
    now
      let m, n be Nat;
A3:   now
        assume m + n < l + 1;
        then m + n <= l by NAT_1:13;
        hence A |^ (m + n) = (A |^ m) ^^ (A |^ n) by A2;
      end;
A4:   now
        assume
A5:     m + n = l + 1;
A6:     now
          assume
A7:       n = 0;
          thus A |^ (m + n) = (A |^ (l + 1)) ^^ {<%>E} by A5,Th13
            .= (A |^ m) ^^ (A |^ n) by A5,A7,Th24;
        end;
A8:     now
          assume that
A9:       m > 0 and
A10:      n > 0;
          consider k such that
A11:      k + 1 = m by A9,NAT_1:6;
          A |^ (m + n) = A |^ (k + n + 1) by A11
            .= (A |^ (k + n)) ^^ A by Th23
            .= A ^^ (A |^ (k + n)) by Th32
            .= A ^^ ((A |^ k) ^^ (A |^ n)) by A2,A5,A11
            .= (A ^^ (A |^ k)) ^^ (A |^ n) by Th18
            .= ((A |^ k) ^^ A) ^^ (A |^ n) by Th32
            .= ((A |^ k) ^^ (A |^ 1)) ^^ (A |^ n) by Th25
            .= (A |^ m) ^^ (A |^ n) by A2,A5,A10,A11,Th2;
          hence A |^ (m + n) = (A |^ m) ^^ (A |^ n);
        end;
        now
          assume
A12:      m = 0;
          thus A |^ (m + n) = {<%>E} ^^ (A |^ (l + 1)) by A5,Th13
            .= (A |^ m) ^^ (A |^ n) by A5,A12,Th24;
        end;
        hence A |^ (m + n) = (A |^ m) ^^ (A |^ n) by A6,A8;
      end;
      assume m + n <= l + 1;
      hence A |^ (m + n) = (A |^ m) ^^ (A |^ n) by A4,A3,XXREAL_0:1;
    end;
    hence P[l + 1];
  end;
A13: P[0]
  proof
    let m, n;
    assume
A14: m + n <= 0;
    then
A15: m = 0;
A16: m + n = 0 by A14;
    hence A |^ (m + n) = (A |^ 0) ^^ {<%>E} by Th13
      .= (A |^ m) ^^ (A |^ n) by A16,A15,Th24;
  end;
  for l holds P[l] from NAT_1:sch 2(A13, A1);
  hence thesis;
end;
