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<b>let </b><font color="Maroon" title="c4">z'</font>, <font color="Maroon" title="c5">x'</font>, <font color="Maroon" title="c6">y'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:65"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c4">z'</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">x'</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">y'</font>
 <b>and </b><br/><a NAME="E2:65"/><i><font color="Green" title="E60">A2</font></i>: 
<font color="Maroon" title="c5">x'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>and </b><br/><a NAME="E3:65"/><i><font color="Green" title="E61">A3</font></i>: 
<font color="Maroon" title="c6">y'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">r0</font>, <font color="Maroon" title="c8">t0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E5:65"/><i><font color="Green" title="E62">A4</font></i>: 
<font color="Maroon" title="c4">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">t0</font>
 <b>and </b><br/><a NAME="E6:65"/><i><font color="Green" title="E63">A5</font></i>: 
<font color="Maroon" title="c7">r0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">x'</font>
 <b>and </b><br/><a NAME="E7:65"/><i><font color="Green" title="E64">A6</font></i>: 
( <font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c6">y'</font> &amp; <font color="Maroon" title="c8">t0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y'</font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:65"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="arytm_2.html#E3:65"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="arytm_3.html#T98" title="ARYTM_3:th.98">ARYTM_3:98</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1"/>
( <font color="Maroon" title="c7">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  or <font color="Maroon" title="c7">r0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1"><b>suppose </b></a><a NAME="E1:65_1_1"/><i><font color="Green" title="E65">A7</font></i>: 
<font color="Maroon" title="c7">r0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ;<br/><div class="add"><b>take </b>
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
;<br/><b>take </b>
<font color="Maroon" title="c8">t0</font>
;<br/><b>thus </b><a NAME="E2:65_1_1"/>
<font color="Maroon" title="c4">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c8">t0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:65"><i><font color="Green" title="E62">A4</font></i></a>, <a class="txt" href="arytm_2.html#E1:65_1_1"><i><font color="Green" title="E65">A7</font></i></a></i>;<br/><a NAME="E3:65_1_1"/>
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">x'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></i>;<br/><b>hence </b><a NAME="E4:65_1_1"/>
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">x'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:65"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/><b>thus </b><a NAME="E5:65_1_1"/>
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">y'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E7:65"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2"><b>suppose </b></a><a NAME="E1:65_1_2"/><i><font color="Green" title="E65">A8</font></i>: 
<font color="Maroon" title="c7">r0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ;<br/><div class="add"><a NAME="E2:65_1_2"/>
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">y'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E7:65"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">t1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E4:65_1_2"/><i><font color="Green" title="E66">A9</font></i>: 
<font color="Maroon" title="c8">t0</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c9">t1</font>
 <b>and </b><br/><a NAME="E5:65_1_2"/><i><font color="Green" title="E67">A10</font></i>: 
<font color="Maroon" title="c9">t1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">y'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T101" title="ARYTM_3:th.101">ARYTM_3:101</a></i>;<br/><a NAME="E6:65_1_2"/>
<font color="Maroon" title="c4">z'</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c9">t1</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c7">r0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:65"><i><font color="Green" title="E62">A4</font></i></a>, <a class="txt" href="arytm_2.html#E4:65_1_2"><i><font color="Green" title="E66">A9</font></i></a>, <a class="ref" href="arytm_3.html#T83" title="ARYTM_3:th.83">ARYTM_3:83</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">t2</font>, <font color="Maroon" title="c11">r1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E8:65_1_2"/><i><font color="Green" title="E68">A11</font></i>: 
<font color="Maroon" title="c4">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t2</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c11">r1</font>
 <b>and </b><br/><a NAME="E9:65_1_2"/><i><font color="Green" title="E69">A12</font></i>: 
<font color="Maroon" title="c10">t2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c9">t1</font>
 <b>and </b><br/><a NAME="E10:65_1_2"/><i><font color="Green" title="E70">A13</font></i>: 
( <font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c7">r0</font> &amp; <font color="Maroon" title="c11">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r0</font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:65_1_2"><i><font color="Green" title="E65">A8</font></i></a>, <a class="ref" href="arytm_3.html#T98" title="ARYTM_3:th.98">ARYTM_3:98</a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">r1</font>
;<br/><b>take </b>
<font color="Maroon" title="c10">t2</font>
;<br/><b>thus </b><a NAME="E11:65_1_2"/>
<font color="Maroon" title="c4">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c10">t2</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8:65_1_2"><i><font color="Green" title="E68">A11</font></i></a></i>;<br/><a NAME="E12:65_1_2"/>
<font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c7">r0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E10:65_1_2"><i><font color="Green" title="E70">A13</font></i></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></i>;<br/><b>hence </b><a NAME="E13:65_1_2"/>
<font color="Maroon" title="c11">r1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">x'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:65"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></i>;<br/><b>thus </b><a NAME="E14:65_1_2"/>
<font color="Maroon" title="c10">t2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">y'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:65_1_2"><i><font color="Green" title="E67">A10</font></i></a>, <a class="txt" href="arytm_2.html#E9:65_1_2"><i><font color="Green" title="E69">A12</font></i></a>, <a class="ref" href="arytm_3.html#T76" title="ARYTM_3:th.76">ARYTM_3:76</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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