<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K7" title="ARYTM_3:func.7">RAT+</a> ;<br/>

<b>given </b><font color="Maroon" title="c2">t</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><a NAME="E2:22"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">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/>

<b>assume </b><a NAME="E3:22"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</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 <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">x</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="E5:22"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T66" title="ARYTM_3:th.66">ARYTM_3:66</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">y</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="E7:22"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T66" title="ARYTM_3:th.66">ARYTM_3:66</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">w</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="E9:22"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c5">w</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T66" title="ARYTM_3:th.66">ARYTM_3:66</a></i>;<br/>
<b>take </b>
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
;<br/>

<b>take </b>
<font color="Maroon" title="c5">w</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c4">y</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c3">x</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c2">t</font>
;<br/>

<a NAME="E10:22"/><i><font color="Green" title="E18">A6</font></i>: 
(  <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="c2">t</font> &amp; <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c2">t</font> )
 
<b>by </b><i><a class="txt" href="quaterni.html#E5:22"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a>, <a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></i>;<br/>
<a NAME="E11:22"/><i><font color="Green" title="E19">A7</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c5">w</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="txt" href="quaterni.html#E7:22"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="quaterni.html#E9:22"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="arytm_3.html#D13" title="ARYTM_3:def.13">ARYTM_3:def 13</a></i>;<br/>
<a NAME="E12:22"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c2">t</font>
 
<b>by </b><i><a class="txt" href="quaterni.html#E10:22"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="arytm_3.html#T74" title="ARYTM_3:th.74">ARYTM_3:74</a></i>;<br/>
<a NAME="E13:22"/>
<font color="Maroon" title="c5">w</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="quaterni.html#E11:22"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="arytm_3.html#T74" title="ARYTM_3:th.74">ARYTM_3:74</a></i>;<br/>
<a NAME="E14:22"/><b>then </b>
<font color="Maroon" title="c5">w</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c2">t</font>
 
<b>by </b><i><a class="txt" href="quaterni.html#E10:22"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="arytm_3.html#T74" title="ARYTM_3:th.74">ARYTM_3:74</a></i>;<br/>
<b>hence </b><a NAME="E15:22"/>
(  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="quaterni.html#E2:22"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="quaterni.html#E3:22"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="quaterni.html#E10:22"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="quaterni.html#E12:22"><i><font color="Green" title="E20">A8</font></i></a></i>;<br/>

<a NAME="E16:22"/><i><font color="Green" title="E21">A9</font></i>: 
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="quaterni.html#E2:22"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="quaterni.html#E5:22"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="arytm_3.html#T56" title="ARYTM_3:th.56">ARYTM_3:56</a></i>;<br/>
<a NAME="E17:22"/><b>then </b><i><font color="Green" title="E22">A10</font></i>: 
 <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="txt" href="quaterni.html#E7:22"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="arytm_3.html#T56" title="ARYTM_3:th.56">ARYTM_3:56</a></i>;<br/>
<a NAME="E18:22"/><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font>
 ;<br/>

<a NAME="E2:22_1"/><b>then </b>
<font color="Maroon" title="c2">t</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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c2">t</font>
 
<b>by </b><i><a class="txt" href="quaterni.html#E5:22"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="arytm_3.html#T56" title="ARYTM_3:th.56">ARYTM_3:56</a></i>;<br/>
<b>hence </b><a NAME="E3:22_1"/>
contradiction
 <b>by </b><i><a class="txt" href="quaterni.html#E2:22"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="arytm_3.html#T69" title="ARYTM_3:th.69">ARYTM_3:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:22"/><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 ;<br/>

<a NAME="E2:22_2"/><b>then </b>
<font color="Maroon" title="c3">x</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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="quaterni.html#E7:22"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="arytm_3.html#T56" title="ARYTM_3:th.56">ARYTM_3:56</a></i>;<br/>
<b>hence </b><a NAME="E3:22_2"/>
contradiction
 <b>by </b><i><a class="txt" href="quaterni.html#E16:22"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="arytm_3.html#T69" title="ARYTM_3:th.69">ARYTM_3:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:22"/>
<font color="Maroon" title="c5">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_3"/>
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<a NAME="E2:22_3"/><b>then </b>
<font color="Maroon" title="c4">y</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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="txt" href="quaterni.html#E9:22"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="arytm_3.html#T56" title="ARYTM_3:th.56">ARYTM_3:56</a></i>;<br/>
<b>hence </b><a NAME="E3:22_3"/>
contradiction
 <b>by </b><i><a class="txt" href="quaterni.html#E17:22"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="arytm_3.html#T69" title="ARYTM_3:th.69">ARYTM_3:69</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:22"/>
(  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">w</font> &amp;  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp;  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp;  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">t</font> &amp; <font color="Maroon" title="c5">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c5">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c5">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">t</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">t</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">t</font> )
 <b>by </b><i><a class="txt" href="quaterni.html#E2:22"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="quaterni.html#E5:22"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="arytm_3.html#T56" title="ARYTM_3:th.56">ARYTM_3:56</a>, <a class="txt" href="quaterni.html#E7:22"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="quaterni.html#E9:22"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="quaterni.html#E18:22"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="quaterni.html#E19:22"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="quaterni.html#E10:22"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="quaterni.html#E11:22"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="quaterni.html#E12:22"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="arytm_3.html#T73" title="ARYTM_3:th.73">ARYTM_3:73</a></i>;<br/>


</div>
