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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">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">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> ;<br/>





<b>assume </b><a NAME="E1:25"/><i><font color="Green">E15</font></i>: 
<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="25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1"/>
( <font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x</font> or  not <font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>suppose </b></a><a NAME="E1:25_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">z</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="txt" href="arytm_1.html#E1:25_1_1"><i><font color="Green">E17</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_1.html#K1" target="_self">-'</a> <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></span>)</span> <a href="arytm_2.html#K7">+</a> <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> = 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="arytm_1.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">z</font></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font>
<b>by </b><i><a class="txt" href="arytm_1.html#E1:25"><i><font color="Green">E15</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="arytm_2.html#K7">+</a> <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></span>)</span></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="arytm_1.html#T6" target="_self">Th6</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_2.html#K7">+</a> <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></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</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"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">y</font></span>)</span> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font></span>)</span> <a href="arytm_2.html#K7">+</a> <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>
<b>by </b><i>, <a class="ref" href="arytm_1.html#T4" target="_self">Th4</a></i>
;<br/><b>hence </b><a NAME="E4:25_1_1"/>
<font color="Maroon">x</font> <a href="arytm_1.html#K1" target="_self">-'</a> <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="hidden.html#R1">=</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> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T12">ARYTM_2:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2"><b>suppose </b></a><a NAME="E1:25_1_2"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">z</font> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:25_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
not <font color="Maroon">z</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/>;<br/><b>thus </b><font color="Maroon">x</font> <a href="arytm_1.html#K1" target="_self">-'</a> <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_3.html#K13">{}</a> 
<b>by </b><i><a class="ref" href="arytm_1.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
<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> <a href="arytm_1.html#K1" target="_self">-'</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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