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<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> ;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( <font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or ( not <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  ) or ( not <font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:21_1_1"/>
<font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T20">ARYTM_2:20</a></i>;<br/><a NAME="E3:21_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:21_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#T3" target="_self">Th3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_2.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_2.html#K7">+</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="arytm_2.html#T7">ARYTM_2:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="arytm_1.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="arytm_1.html#T4" target="_self">Th4</a>, </i>
;<br/><b>hence </b><a NAME="E6:21_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font> = 
<font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="arytm_1.html#E3"><i><font color="Green">Lemma14</font></i></a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_2.html#D8">ARYTM_2:def 8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:21_1_3"/><i><font color="Green">E41</font></i>: 
not <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E2:21_1_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_3_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:21_1_3_1"/>
<font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:21_1_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="arytm_1.html#T4" target="_self">Th4</a>, <a class="ref" href="arytm_1.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div><a NAME="E4:21_1_3"/>
<font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">y</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T20">ARYTM_2:20</a></i>;<br/><a NAME="E5:21_1_3"/><b>then </b><i><font color="Green">E45</font></i>: 
not <font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font> = 
 <a href="arytm_3.html#K13">{}</a> 
<b>by </b><i><a class="ref" href="arytm_1.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
<font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_1.html#T6" target="_self">Th6</a>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:21_1_4"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E2:21_1_4"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E3:21_1_4"/>
not <font color="Maroon">x</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="arytm_1.html#T4" target="_self">Th4</a>, <a class="ref" href="arytm_1.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font> = 
 <a href="arytm_3.html#K13">{}</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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