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

<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">t</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>thus </b><a NAME="E1:101"/>
( <font color="Maroon">s</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t</font> implies <font color="Maroon">s</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">z</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><a NAME="E1:101_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">z</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="arytm_3.html#D13" target="_self">ARYTM_3:def 13</a><br/>

<b>take </b>
<font color="Maroon">z</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="arytm_3.html#D13" target="_self">ARYTM_3:def 13</a><br/>

<a NAME="E2:101_1"/>
<font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">z</font></span>)</span> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T32" target="_self">Th32</a>, <a class="ref" href="arytm_3.html#T17" target="_self">Th17</a></i>;<br/>
<b>hence </b><a NAME="E3:101_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T24" target="_self">Th24</a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">z</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><a NAME="E2:101"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">z</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="arytm_3.html#D13" target="_self">ARYTM_3:def 13</a><br/>

<b>take </b>
<font color="Maroon">z</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="arytm_3.html#D13" target="_self">ARYTM_3:def 13</a><br/>

<b>thus </b><a NAME="E3:101"/>
<font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t</font></span>)</span> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T32" target="_self">Th32</a>, <a class="ref" href="arytm_3.html#T17" target="_self">Th17</a></i>;<br/>


</div>
