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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="19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1"/>
( <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font> or  not <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="19_1_1"><b>suppose </b></a><a NAME="E1:19_1_1"/>
<font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:19_1_1"/><b>then </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_2.html#K7">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:19_1_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> iff <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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>suppose </b></a><a NAME="E1:19_1_2"/><i><font color="Green">E15</font></i>: 
not <font color="Maroon">y</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:19_1_2"/><b>then </b><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="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E3:19_1_2"/>
<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/><b>hence </b><a NAME="E4:19_1_2"/>
( <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> iff <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="ref" href="arytm_1.html#T3" target="_self">Th3</a>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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