<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies ( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> ) &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> ) )</span><br/>

<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E41">A1</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> ) &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E2:54"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> ) &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 </span><b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1_1"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> = 
<font color="Maroon" title="c2">y</font>
<b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_1_1_1_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_1_2"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> = 
<font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_1_1_1_2"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_3"><b>suppose </b></a><a NAME="E1:54_1_1_1_3"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_4"><b>suppose </b></a><a NAME="E1:54_1_1_1_4"/><i><font color="Green" title="E42">A4</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_1_1_1_4"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_5"><b>suppose </b></a><a NAME="E1:54_1_1_1_5"/><i><font color="Green" title="E42">A5</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
<br/>.= 
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_1_1_1_5"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_6"><b>suppose </b></a><a NAME="E1:54_1_1_1_6"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_1_6"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:54_1"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:54"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="mod_2.html#E62"><i><font color="Green" title="E37">Lm7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> ) &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E4:54"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> ) &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E5:54"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><a href="mod_2.html#K10" title="MOD_2:func.10">-</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="mod_2.html#E62"><i><font color="Green" title="E37">Lm7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> ) &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E6:54"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> ) &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_2_1_1"/>
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 </span><b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1"><b>suppose </b></a><a NAME="E1:54_2_1_1_1"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_2_1_1_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_2"><b>suppose </b></a><a NAME="E1:54_2_1_1_2"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_2_1_1_2"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_3"><b>suppose </b></a><a NAME="E1:54_2_1_1_3"/><i><font color="Green" title="E42">A8</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> = 
<font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_2_1_1_3"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_4"><b>suppose </b></a><a NAME="E1:54_2_1_1_4"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font></span><br/><div class="add"><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> = 
<font color="Maroon" title="c2">y</font>
<b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_2_1_1_4"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_5"><b>suppose </b></a><a NAME="E1:54_2_1_1_5"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:54_2_1_1_5"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:54_2"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:54"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">b</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> ) &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_3_1_1"/>
( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 </span><b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_1_1"><b>suppose </b></a><a NAME="E1:54_3_1_1_1"/><i><font color="Green" title="E42">A10</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_3_1_1_1"/><b>then </b><i><font color="Green" title="E43">A11</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_3_1_1_1"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E2:54_3_1_1_1"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_1_2"><b>suppose </b></a><a NAME="E1:54_3_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_3_1_1_2"/><b>then </b><i><font color="Green" title="E42">A12</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E2:54_3_1_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_1_3"><b>suppose </b></a><a NAME="E1:54_3_1_1_3"/><i><font color="Green" title="E42">A13</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_3_1_1_3"/><b>then </b>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_3_1_1_3"><i><font color="Green" title="E42">A13</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_1_4"><b>suppose </b></a><a NAME="E1:54_3_1_1_4"/><i><font color="Green" title="E42">A14</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_3_1_1_4"/><b>then </b>
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E3:54_3_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54_3_1_1_4"><i><font color="Green" title="E42">A14</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_1_5"><b>suppose </b></a><a NAME="E1:54_3_1_1_5"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_3_1_1_5"/><b>then </b>
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> )
 <b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E3:54_3_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_1_6"><b>suppose </b></a><a NAME="E1:54_3_1_1_6"/><i><font color="Green" title="E42">A15</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> = 
<font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font>
<b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_3_1_1_6"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_1_7"><b>suppose </b></a><a NAME="E1:54_3_1_1_7"/><i><font color="Green" title="E42">A16</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_3_1_1_7"/><b>then </b><i><font color="Green" title="E43">A17</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> = 
<font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_3_1_1_7"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E2:54_3_1_1_7"><i><font color="Green" title="E43">A17</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:54_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E8:54"/>
<font color="Maroon" title="c5">b</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> ) &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E9:54"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_4_1_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_1_1_1"><b>case </b></a><a NAME="E1:54_4_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )</span><br/><div class="add"><b>hence </b><a NAME="E2:54_4_1_1_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54"><i><font color="Green" title="E41">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_1_1_2"><b>case </b></a><a NAME="E1:54_4_1_1_2"/><i><font color="Green" title="E42">A18</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <br/>( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c6">y</font> = 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><b>take </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )</span><br/><a NAME="E3:54_4_1_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54_4_1_1_2"><i><font color="Green" title="E42">A18</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E4:54_4_1_1_2"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54"><i><font color="Green" title="E41">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_1_1_3"><b>case </b></a><a NAME="E1:54_4_1_1_3"/><i><font color="Green" title="E42">A19</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <br/>( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c6">y</font> = 2 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><b>take </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )</span><br/><a NAME="E3:54_4_1_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54_4_1_1_3"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E4:54_4_1_1_3"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54"><i><font color="Green" title="E41">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:54_4"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">a</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> st <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E10:54"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54"><i><font color="Green" title="E41">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/>

<b>thus </b><a NAME="E11:54"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_5_1_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 </span><b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_1"><b>suppose </b></a><a NAME="E1:54_5_1_1_1"/><i><font color="Green" title="E42">A20</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_5_1_1_1"/><b>then </b><i><font color="Green" title="E43">A21</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>thus </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_5_1_1_1"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E2:54_5_1_1_1"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_2"><b>suppose </b></a><a NAME="E1:54_5_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_5_1_1_2"/><b>then </b>
( <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E3:54_5_1_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_3"><b>suppose </b></a><a NAME="E1:54_5_1_1_3"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_5_1_1_3"/><b>then </b>
( <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E3:54_5_1_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_4"><b>suppose </b></a><a NAME="E1:54_5_1_1_4"/><i><font color="Green" title="E42">A22</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_5_1_1_4"/><b>then </b>
( <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E3:54_5_1_1_4"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54_5_1_1_4"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_5"><b>suppose </b></a><a NAME="E1:54_5_1_1_5"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_5_1_1_5"/><b>then </b><i><font color="Green" title="E42">A23</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E2:54_5_1_1_5"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_6"><b>suppose </b></a><a NAME="E1:54_5_1_1_6"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_5_1_1_6"/><b>then </b><i><font color="Green" title="E42">A24</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E2:54_5_1_1_6"><i><font color="Green" title="E42">A24</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_7"><b>suppose </b></a><a NAME="E1:54_5_1_1_7"/><i><font color="Green" title="E42">A25</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_5_1_1_7"/><b>then </b>
( <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E3:54_5_1_1_7"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="mod_2.html#E1:54_5_1_1_7"><i><font color="Green" title="E42">A25</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_8"><b>suppose </b></a><a NAME="E1:54_5_1_1_8"/><i><font color="Green" title="E42">A26</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_5_1_1_8"/><b>then </b><i><font color="Green" title="E43">A27</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> = 
1
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_5_1_1_8"><i><font color="Green" title="E42">A26</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E2:54_5_1_1_8"><i><font color="Green" title="E43">A27</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_9"><b>suppose </b></a><a NAME="E1:54_5_1_1_9"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_5_1_1_9"/><b>then </b><i><font color="Green" title="E42">A28</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E2:54_5_1_1_9"><i><font color="Green" title="E42">A28</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_10"><b>suppose </b></a><a NAME="E1:54_5_1_1_10"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_5_1_1_10"/><b>then </b><i><font color="Green" title="E42">A29</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E2:54_5_1_1_10"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_11"><b>suppose </b></a><a NAME="E1:54_5_1_1_11"/><i><font color="Green" title="E42">A30</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span></span><br/><div class="add"><a NAME="E2:54_5_1_1_11"/><b>then </b><i><font color="Green" title="E43">A31</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>;<br/><b>hence </b><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> = 
2
<b>by </b><i><a class="txt" href="mod_2.html#E1:54_5_1_1_11"><i><font color="Green" title="E42">A30</font></i></a>, <a class="ref" href="mod_2.html#D17" target="_self" title="MOD_2:def.17">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_2.html#E2:54_5_1_1_11"><i><font color="Green" title="E43">A31</font></i></a>, <a class="ref" href="mod_2.html#D16" target="_self" title="MOD_2:def.16">Def16</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:54_5"/>
<font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="mod_2.html#K11" title="MOD_2:func.11">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mod_2.html#K12" title="MOD_2:func.12">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div>
