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

<span class="kw">let </span><font color="Maroon" title="c3">t</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">t</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>  implies  ex <font color="Olive" title="b1">r</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="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; not <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:94"/><span class="lab"><font color="Green" title="E76">A1</font></span>: 
<font color="Maroon" title="c3">t</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</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="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; not <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )</span><br/>

<a NAME="E2:94"/><span class="lab"><font color="Green" title="E77">A2</font></span>: 
1 <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T48" title="ORDINAL2:th.48">ORDINAL2:48</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:94_1"/>
(  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c3">t</font> or <font color="Maroon" title="c3">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_1"><span class="kw">suppose </span></a><a NAME="E1:94_1_1"/><span class="lab"><font color="Green" title="E78">A3</font></span>: 
 <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c3">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</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="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; not <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c4">r</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> <span class="kw"> such that </span><br/><a NAME="E3:94_1_1"/><span class="lab"><font color="Green" title="E79">A4</font></span>: 
 <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T66" target="_self" title="ARYTM_3:th.66">Th66</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c4">r</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; not <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )</span><br/><a NAME="E4:94_1_1"/>
( <font color="Maroon" title="c4">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  &amp; <font color="Maroon" title="c4">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="txt" href="#E3:94_1_1"><span class="lab"><font color="Green" title="E79">A4</font></span></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#T64" target="_self" title="ARYTM_3:th.64">Th64</a></span>;<br/><a NAME="E5:94_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T73" target="_self" title="ARYTM_3:th.73">Th73</a></span>;<br/><span class="kw">hence </span><a NAME="E6:94_1_1"/>
<font color="Maroon" title="c4">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c3">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_1_1"><span class="lab"><font color="Green" title="E78">A3</font></span></a>, <a class="ref" href="arytm_3.html#T74" target="_self" title="ARYTM_3:th.74">Th74</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><br/><a NAME="E7:94_1_1"/><span class="lab"><font color="Green" title="E80">A5</font></span>: 
1 <a href="ordinal3.html#K9" title="ORDINAL3:func.9">*^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></span>;<br/><span class="kw">assume </span><a NAME="E8:94_1_1"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E9:94_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A6</font></span>: 
 <a href="arytm_3.html#K7" title="ARYTM_3:func.7">denominator</a> <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#D9" target="_self" title="ARYTM_3:def.9">Def9</a></span>;<br/><a NAME="E10:94_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c4">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">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T56" title="ORDINAL2:th.56">ORDINAL2:56</a></span>;<br/><a NAME="E11:94_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A7</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="ordinal3.html#K8" title="ORDINAL3:func.8">+^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c4">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94_1_1"><span class="lab"><font color="Green" title="E79">A4</font></span></a>, <a class="txt" href="#E9:94_1_1"><span class="lab"><font color="Green" title="E81">A6</font></span></a>, <a class="txt" href="#E7:94_1_1"><span class="lab"><font color="Green" title="E80">A5</font></span></a>, <a class="ref" href="arytm_3.html#T46" target="_self" title="ARYTM_3:th.46">Th46</a></span>;<br/><a NAME="E12:94_1_1"/><span class="kw">then </span>
 <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c4">r</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T27" title="ORDINAL3:th.27">ORDINAL3:27</a></span>;<br/><a NAME="E13:94_1_1"/><span class="kw">then </span>
(  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  or  <a href="arytm_3.html#K6" title="ARYTM_3:func.6">numerator</a> <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T19" title="ORDINAL3:th.19">ORDINAL3:19</a></span>;<br/><span class="kw">hence </span><a NAME="E14:94_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="txt" href="#E11:94_1_1"><span class="lab"><font color="Green" title="E82">A7</font></span></a>, <a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="94_1_2"><span class="kw">suppose </span></a><a NAME="E1:94_1_2"/><span class="lab"><font color="Green" title="E78">A8</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</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="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; not <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c4">r</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> <span class="kw"> such that </span><br/><a NAME="E3:94_1_2"/><span class="lab"><font color="Green" title="E79">A9</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T66" target="_self" title="ARYTM_3:th.66">Th66</a></span>;<br/><a NAME="E4:94_1_2"/><span class="lab"><font color="Green" title="E80">A10</font></span>: 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T71" target="_self" title="ARYTM_3:th.71">Th71</a></span>;<br/><a NAME="E5:94_1_2"/>
<font color="Maroon" title="c4">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94"><span class="lab"><font color="Green" title="E76">A1</font></span></a>, <a class="txt" href="#E3:94_1_2"><span class="lab"><font color="Green" title="E79">A9</font></span></a>, <a class="ref" href="arytm_3.html#T56" target="_self" title="ARYTM_3:th.56">Th56</a></span>;<br/><a NAME="E6:94_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A11</font></span>: 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:94_1_2"><span class="lab"><font color="Green" title="E80">A10</font></span></a>, <a class="ref" href="arytm_3.html#T73" target="_self" title="ARYTM_3:th.73">Th73</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c4">r</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c3">t</font> &amp; not <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )</span><br/><a NAME="E7:94_1_2"/><span class="lab"><font color="Green" title="E82">A12</font></span>: 
1 <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#K9" title="ARYTM_3:func.9">+</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T56" target="_self" title="ARYTM_3:th.56">Th56</a></span>;<br/><a NAME="E8:94_1_2"/><span class="lab"><font color="Green" title="E83">A13</font></span>: 
<font color="Maroon" title="c4">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c3">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94_1_2"><span class="lab"><font color="Green" title="E79">A9</font></span></a>, <a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">Def13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:94_1_2_1"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:94_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="c3">t</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c3">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:94_1_2"><span class="lab"><font color="Green" title="E79">A9</font></span></a>, <a class="ref" href="arytm_3.html#T56" target="_self" title="ARYTM_3:th.56">Th56</a></span>;<br/><span class="kw">hence </span><a NAME="E3:94_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94"><span class="lab"><font color="Green" title="E76">A1</font></span></a>, <a class="ref" href="arytm_3.html#T69" target="_self" title="ARYTM_3:th.69">Th69</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:94_1_2"/>
<font color="Maroon" title="c4">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c3">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:94_1_2"><span class="lab"><font color="Green" title="E83">A13</font></span></a>, <a class="ref" href="arytm_3.html#T73" target="_self" title="ARYTM_3:th.73">Th73</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> </span><br/><a NAME="E11:94_1_2"/>
<font color="Maroon" title="c4">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a>  <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:94_1_2"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="txt" href="#E8:94_1_2"><span class="lab"><font color="Green" title="E83">A13</font></span></a>, <a class="ref" href="arytm_3.html#T74" target="_self" title="ARYTM_3:th.74">Th74</a></span>;<br/><span class="kw">hence </span><a NAME="E12:94_1_2"/>
not <font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:94_1_2"><span class="lab"><font color="Green" title="E81">A11</font></span></a>, <a class="txt" href="#E7:94_1_2"><span class="lab"><font color="Green" title="E82">A12</font></span></a>, <a class="ref" href="arytm_3.html#T79" target="_self" title="ARYTM_3:th.79">Th79</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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