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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#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  implies <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )</span><br/>



<b>set </b><font color="Maroon" title="c5">nr</font> =  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">r</font>;<br/>
<b>set </b><font color="Maroon" title="c6">dr</font> =  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">r</font>;<br/>
<b>set </b><font color="Maroon" title="c7">ns</font> =  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c4">s</font>;<br/>
<b>set </b><font color="Maroon" title="c8">ds</font> =  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c4">s</font>;<br/>
<a NAME="E1:80"/><i><font color="Green" title="E71">A1</font></i>: 
(  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">r</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>  &amp;  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c4">s</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>  &amp;  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">s</font></span>)</span> <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>by </b><i><a class="ref" href="arytm_3.html#T41" target="_self" title="ARYTM_3:th.41">Th41</a></i>;<br/>
<a NAME="E2:80"/><b>then </b><i><font color="Green" title="E72">A2</font></i>: 
<span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c4">s</font></span>)</span> <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>by </b><i><a class="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</a></i>;<br/>
<a NAME="E3:80"/><i><font color="Green" title="E73">A3</font></i>: 
(  <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</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="ref" href="arytm_3.html#D8" target="_self" title="ARYTM_3:def.8">Def8</a>, <a class="ref" href="arytm_3.html#D9" target="_self" title="ARYTM_3:def.9">Def9</a>, <a class="txt" href="arytm_3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E4:80"/>
<font color="Maroon" title="c3">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span><br/>

<a NAME="E5:80"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="arytm_3.html#K8" title="ARYTM_3:func.8">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <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#K8" title="ARYTM_3:func.8">/</a> 1
 
<b>by </b><i><a class="txt" href="arytm_3.html#E3:80"><i><font color="Green" title="E73">A3</font></i></a>, <a class="ref" href="arytm_3.html#T45" target="_self" title="ARYTM_3:th.45">Th45</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p3">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
<b>by </b><i><a class="txt" href="arytm_3.html#E2:80"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="arytm_3.html#T51" target="_self" title="ARYTM_3:th.51">Th51</a></i>
<br/>.= 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
<b>by </b><i><a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</a></i>
;<br/>
<a NAME="E7:80"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <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="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</a></i>;<br/>
<a NAME="E8:80"/><b>then </b>
<span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c4">s</font></span>)</span> <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="ref" href="ordinal3.html#T29" title="ORDINAL3:th.29">ORDINAL3:29</a></i>;<br/>
<a NAME="E9:80"/><b>then </b>
 <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c3">r</font> <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="arytm_3.html#E1:80"><i><font color="Green" title="E71">A1</font></i></a>, <a class="ref" href="ordinal3.html#T34" title="ORDINAL3:th.34">ORDINAL3:34</a></i>;<br/>
<b>hence </b><a NAME="E10:80"/>
<font color="Maroon" title="c3">r</font> <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="ref" href="arytm_3.html#T43" target="_self" title="ARYTM_3:th.43">Th43</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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