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<b>let </b><font color="Maroon" title="c3">r</font>, <font color="Maroon" title="c4">s</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">s</font> implies  ex <font color="Olive" title="b1">t</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>  st <br/>( <font color="Maroon" title="c3">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">s</font> ) )</span><br/>



<b>assume </b><a NAME="E1:106"/><i><font color="Green" title="E81">A1</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t</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>  st <br/>( <font color="Maroon" title="c3">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">s</font> )</span><br/>

<b>then 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#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E3:106"/><i><font color="Green" title="E82">A2</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">Def13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">y</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="E5:106"/><i><font color="Green" title="E83">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">y</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><font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c3">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c7">t</font> &amp; <font color="Maroon" title="c7">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">s</font> )</span><br/>

<a NAME="E6:106"/><i><font color="Green" title="E84">A4</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="txt" href="#E3:106"><i><font color="Green" title="E82">A2</font></i></a>, <a class="txt" href="#E5:106"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="arytm_3.html#T57" target="_self" title="ARYTM_3:th.57">Th57</a></i>;<br/>
<a NAME="E7:106"/><i><font color="Green" title="E85">A5</font></i>: 
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c7">t</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> &amp; <font color="Maroon" title="c3">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> &amp; <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )
 
<b>by </b><i><a class="txt" href="#E1:106"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="arytm_3.html#T56" target="_self" title="ARYTM_3:th.56">Th56</a></i>;<br/>
<a NAME="E8:106"/>
( <font color="Maroon" title="c3">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c7">t</font> &amp; <font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">t</font> )
 
<b>by </b><i><a class="txt" href="#E1:106"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="#E6:106"><i><font color="Green" title="E84">A4</font></i></a>, <a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E9:106"/>
<font color="Maroon" title="c3">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c7">t</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T73" target="_self" title="ARYTM_3:th.73">Th73</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">s</font></span><br/>

<a NAME="E10:106"/>
( <font color="Maroon" title="c7">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c4">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">t</font> )
 
<b>by </b><i><a class="txt" href="#E6:106"><i><font color="Green" title="E84">A4</font></i></a>, <a class="txt" href="#E7:106"><i><font color="Green" title="E85">A5</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#T69" target="_self" title="ARYTM_3:th.69">Th69</a></i>;<br/>
<b>hence </b><a NAME="E11:106"/>
<font color="Maroon" title="c7">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">s</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T73" target="_self" title="ARYTM_3:th.73">Th73</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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