:: Basic Properties of Real Numbers - Requirements :: by Library Committee :: :: Received February 27, 2003 :: Copyright (c) 2003-2018 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 XXREAL_0, NUMBERS, ARYTM_2, SUBSET_1, ARYTM_3, ZFMISC_1, CARD_1, XCMPLX_0, REAL_1; notations TARSKI, ZFMISC_1, SUBSET_1, ARYTM_2, ORDINAL1, NUMBERS, XXREAL_0, XREAL_0; constructors ARYTM_2, NUMBERS, XXREAL_0, XREAL_0; registrations ARYTM_2, XREAL_0, ORDINAL1; requirements SUBSET, BOOLE; begin :: This file contains statements which are obvious for Mizar checker if :: "requirements REAL" is included in the environment description of an article. :: They are published for testing purposes only. :: Users should use appropriate requirements instead of referencing :: to these theorems. :: Note that the checker needs also "requirements BOOLE" to accept :: the statements with attribute 'zero'. reserve x, y, z for Real; theorem :: REAL:1 x <= y & x is positive implies y is positive; theorem :: REAL:2 x <= y & y is negative implies x is negative; theorem :: REAL:3 x <= y & x is non negative implies y is non negative; theorem :: REAL:4 x <= y & y is non positive implies x is non positive; theorem :: REAL:5 x <= y & y is non zero & x is non negative implies y is positive; theorem :: REAL:6 x <= y & x is non zero & y is non positive implies x is negative; theorem :: REAL:7 not x <= y & x is non positive implies y is negative; theorem :: REAL:8 not x <= y & y is non negative implies x is positive; ::theorem :: numeral(X) & X <> 0 implies X is positive;