:: Field Properties of Complex Numbers - Requirements
::  by Library Committee
::
:: Received May 29, 2003
:: Copyright (c) 2003 Association of Mizar Users

environ

 vocabularies XCMPLX_0, ORDINAL1, ARYTM_3, CARD_1, NUMBERS, SUBSET_1, ARYTM_0,
      ARYTM_1, RELAT_1;
 notations SUBSET_1, ORDINAL1, NUMBERS, ARYTM_0, XCMPLX_0;
 constructors FUNCT_4, ARYTM_0, XCMPLX_0;
 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 number;

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;