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

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> ;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z</font></span>)</span> = 
<font color="Maroon" title="c4">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="txt" href="arytm_2.html#E1:39_1_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z</font></span>)</span> = 
<font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="txt" href="arytm_2.html#E1:39_1_2"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_3"><b>suppose </b></a><a NAME="E1:39_1_3"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z</font></span>)</span> = 
<font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="txt" href="arytm_2.html#E1:39_1_3"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:39_1_4"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>and </b><br/><a NAME="E2:39_1_4"/><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>and </b><br/><a NAME="E3:39_1_4"/><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><div class="add"><div><i><font color="Green" title="E42">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_4_1"/>
 <a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><a NAME="E2:39_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E21"><i><font color="Green" title="E16">Lm12</font></i></a></i>;<br/><a NAME="E3:39_1_4_1"/><b>then </b>
(  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  )
 <b>by </b><i><a class="txt" href="arytm_2.html#E45"><i><font color="Green" title="E37">Lm28</font></i></a></i>;<br/><b>hence </b><a NAME="E4:39_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:39_1_4"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="arytm_2.html#E3:39_1_4"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="arytm_2.html#E20"><i><font color="Green" title="E15">Lm11</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E43">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_4_2"/>
 <a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 ;<br/><a NAME="E2:39_1_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E21"><i><font color="Green" title="E16">Lm12</font></i></a></i>;<br/><a NAME="E3:39_1_4_2"/><b>then </b>
(  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  or  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  )
 <b>by </b><i><a class="txt" href="arytm_2.html#E45"><i><font color="Green" title="E37">Lm28</font></i></a></i>;<br/><b>hence </b><a NAME="E4:39_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:39_1_4"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="arytm_2.html#E2:39_1_4"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="arytm_2.html#E20"><i><font color="Green" title="E15">Lm11</font></i></a></i>;<br/></div><b>end;</b></div><b>thus </b><font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z</font></span>)</span> = 
<font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p3">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E2:39_1_4"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="arytm_2.html#E3:39_1_4"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>
<br/>.= 
 <a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <span class="p3">(<span class="default"><a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p5">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E1:39_1_4"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="arytm_2.html#E4:39_1_4"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>
<br/>.= 
 <a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p3">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E22"><i><font color="Green" title="E17">Lm13</font></i></a></i>
<br/>.= 
 <a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p3">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E44"><i><font color="Green" title="E36">Lm27</font></i></a></i>
<br/>.= 
 <a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <span class="p3">(<span class="default"><a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p5">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p2">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c5">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E22"><i><font color="Green" title="E17">Lm13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_2.html#K4" title="ARYTM_2:func.4">GLUED</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="arytm_2.html#K5" title="ARYTM_2:func.5">+</a> <span class="p3">(<span class="default"><a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="txt" href="arytm_2.html#E3:39_1_4"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="arytm_2.html#E5:39_1_4"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z</font>
<b>by </b><i><a class="txt" href="arytm_2.html#E1:39_1_4"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="arytm_2.html#E2:39_1_4"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="arytm_2.html#D8" target="_self" title="ARYTM_2:def.8">Def8</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
