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<b>let </b><font color="Maroon" title="c3">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="c4">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="E2:100"/><i><font color="Green" title="E81">A1</font></i>: 
( <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">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="E3:100"/><i><font color="Green" title="E82">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="c3">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="c3">A</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">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="E5:100"/><i><font color="Green" title="E83">A3</font></i>: 
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T66" target="_self" title="ARYTM_3:th.66">Th66</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">x</font>
;<br/>

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

<a NAME="E6:100"/>
(  <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="c4">t</font> &amp; <font color="Maroon" title="c5">x</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c4">t</font> )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E5:100"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">Def13</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="E7:100"/>
(  <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="c3">A</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:100"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="arytm_3.html#E3:100"><i><font color="Green" title="E82">A2</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:100"/>
 <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">x</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:100"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="arytm_3.html#E5:100"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="arytm_3.html#T56" target="_self" title="ARYTM_3:th.56">Th56</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:100_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font>
 ;<br/><a NAME="E2:100_1"/><b>then </b>
<font color="Maroon" title="c4">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="c4">t</font> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">+</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E5:100"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="arytm_3.html#T56" target="_self" title="ARYTM_3:th.56">Th56</a></i>;<br/><b>hence </b><a NAME="E3:100_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:100"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="arytm_3.html#T69" target="_self" title="ARYTM_3:th.69">Th69</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E10:100"/>
<font color="Maroon" title="c4">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> 
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:100"><i><font color="Green" title="E81">A1</font></i></a></i>;<br/>


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