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





<b>assume </b><a NAME="E1:52"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font>
 ;<br/>

<b>consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><br/><a NAME="E3:52"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">z</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1"/>
( <font color="Maroon">z</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>suppose </b></a><a NAME="E1:52_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">z</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E16"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/><a NAME="E3:52_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:52_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E17"><i><font color="Green">Lemma53</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>suppose </b></a><a NAME="E1:52_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">y</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:52_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">z</font></span>)</span> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E16"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/><a NAME="E3:52_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:52_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E17"><i><font color="Green">Lemma53</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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