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<b>let </b><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> ;<br/>





<b>assume </b><a NAME="E1:114"/><i><font color="Green" title="E88">A1</font></i>: 
( <font color="Maroon" title="c3">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c5">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</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" title="c4">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c3">s</font> or <font color="Maroon" title="c3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c4">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" title="c4">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c3">s</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">t0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> <b> such that </b><br/><a NAME="E3:114_1_1"/><i><font color="Green" title="E89">A2</font></i>: 
( <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c6">t0</font> &amp; <font color="Maroon" title="c6">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:114"><i><font color="Green" title="E88">A1</font></i></a>, <a class="ref" href="arytm_3.html#T94" target="_self" title="ARYTM_3:th.94">Th94</a></i>;<br/><b>take </b>
<font color="Maroon" title="c4">r</font>
;<br/><b>take </b>
<font color="Maroon" title="c6">t0</font>
;<br/><b>thus </b><a NAME="E4:114_1_1"/>
( <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c6">t0</font> &amp; <font color="Maroon" title="c4">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c6">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c6">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">t</font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:114"><i><font color="Green" title="E88">A1</font></i></a>, <a class="txt" href="arytm_3.html#E3:114_1_1"><i><font color="Green" title="E89">A2</font></i></a></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" title="E89">A3</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c4">r</font>
 ;<br/><div class="add"><a NAME="E2:114_1_2"/>
( <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  &amp;  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font> )
 <b>by </b><i><a class="ref" href="arytm_3.html#T56" target="_self" title="ARYTM_3:th.56">Th56</a>, <a class="ref" href="arytm_3.html#T71" target="_self" title="ARYTM_3:th.71">Th71</a></i>;<br/><b>hence </b><a NAME="E3:114_1_2"/>
 ex <font color="Olive" title="b1">r0</font>, <font color="Olive" title="b2">t0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a>  st <br/>( <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r0</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Olive" title="b2">t0</font> &amp; <font color="Olive" title="b1">r0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c4">r</font> &amp; <font color="Olive" title="b2">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b2">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">t</font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:114"><i><font color="Green" title="E88">A1</font></i></a>, <a class="txt" href="arytm_3.html#E1:114_1_2"><i><font color="Green" title="E89">A3</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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