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<b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> ;<br/><b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> ;<br/><b>let </b><font color="Maroon">t</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> ;<br/>





<b>assume </b><a NAME="E1:114"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:114_1"/>
( <font color="Maroon">r</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">s</font> or <font color="Maroon">s</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_1"><b>suppose </b></a><a NAME="E1:114_1_1"/>
<font color="Maroon">r</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">t0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> <b> such that </b><br/><a NAME="E3:114_1_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t0</font> &amp; <font color="Maroon">t0</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">t</font> )
 <b>by </b><i>, <a class="txt" href="arytm_3.html#E45"><i><font color="Green">Lemma86</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">r</font>
;<br/><b>take </b>
<font color="Maroon">t0</font>
;<br/><b>thus </b><a NAME="E4:114_1_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t0</font> &amp; <font color="Maroon">r</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">r</font> &amp; <font color="Maroon">t0</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="114_1_2"><b>suppose </b></a><a NAME="E1:114_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">s</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:114_1_2"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="arytm_3.html#K11" target="_self">+</a> <a href="arytm_3.html#K13" target="_self">{}</a>  &amp;  <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">t</font> )
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E3:114_1_2"/>
 ex <font color="Olive">r0</font>, <font color="Olive">t0</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a>  st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">r0</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Olive">t0</font> &amp; <font color="Olive">r0</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">r</font> &amp; <font color="Olive">t0</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">t</font> &amp; <font color="Olive">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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