:: Field Properties of Complex Numbers - Requirements
::  by Library Committee
::
:: Received May 29, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users
::           (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.

environ

 vocabularies XCMPLX_0, ARYTM_3, CARD_1, NUMBERS, SUBSET_1, ARYTM_0, ARYTM_1,
      RELAT_1;
 notations TARSKI, SUBSET_1, ORDINAL1, NUMBERS, ARYTM_0, XCMPLX_0;
 constructors FUNCT_4, ARYTM_0, XCMPLX_0, NUMBERS;
 registrations XCMPLX_0, ORDINAL1;
 requirements NUMERALS, SUBSET, BOOLE;


begin

:: This file contains statements which are obvious for Mizar checker if
:: "requirements ARITHM" is included in the environment description
:: of an article. "requirements NUMERALS" is also required.
:: They are published for testing purposes only.
:: Users should use appropriate requirements instead of referencing
:: to these theorems.
:: Some of these items need also other requirements for proper work.

reserve x for Complex;

theorem :: ARITHM:1
  x + 0 = x;

theorem :: ARITHM:2
  x * 0 = 0;

theorem :: ARITHM:3
  1 * x = x;

theorem :: ARITHM:4
  x - 0 = x;

theorem :: ARITHM:5
  0 / x = 0;

theorem :: ARITHM:6
  x / 1 = x;