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<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/><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>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green">E15</font></i>: 
not <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green">E17</font></i>: 
not <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font>
 ;<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">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">z</font> or  not <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">z</font> )
 </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"/>
<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">z</font>
 ;<br/><div class="add"><a NAME="E2:39_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="arytm_1.html#E2:39"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E3:39_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="arytm_1.html#K2" target="_self">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="arytm_1.html#K1" target="_self">-'</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="arytm_1.html#K2" target="_self">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font></span>)</span> <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">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font></span>)</span> )
 <b>by </b><i><a class="ref" href="arytm_1.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E4:39_1_1"/>
<font color="Maroon">x</font> <a href="arytm_1.html#K2" target="_self">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="arytm_1.html#K2" target="_self">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_1.html#E11"><i><font color="Green">Lemma27</font></i></a>, , <a class="ref" href="arytm_1.html#T7" target="_self">Th7</a></i>;<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"/>
not <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">z</font>
 ;<br/><div class="add"><a NAME="E2:39_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
( not <font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="arytm_1.html#E2:39"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E3:39_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/><b>hence </b><font color="Maroon">x</font> <a href="arytm_1.html#K2" target="_self">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="arytm_3.html#K13">{}</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="arytm_1.html#D1" target="_self">Def1</a>, <a class="ref" href="arytm_1.html#T8" target="_self">Th8</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="arytm_1.html#K2" target="_self">-</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_1.html#D1" target="_self">Def1</a>, <a class="ref" href="arytm_1.html#T8" target="_self">Th8</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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