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<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> ;<br/>

<b>assume </b><a NAME="E1:98"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 ;<br/>

<a NAME="E2:98"/><i><font color="Green">E83</font></i>: 
1 <a href="arytm_3.html#K2" target="_self">+^</a> 1 <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> 1
 
<b>by </b><i><a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a></i>;<br/>
<a NAME="E3:98"/><b>then </b><i><font color="Green">E84</font></i>: 
1 <a href="arytm_3.html#K2" target="_self">+^</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_1"/>
(  <a href="arytm_3.html#K14" target="_self">one</a>  <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14" target="_self">one</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_1"><b>suppose </b></a><a NAME="E1:98_1_1"/><i><font color="Green">E85</font></i>: 
 <a href="arytm_3.html#K14" target="_self">one</a>  <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>4</sub></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><br/><a NAME="E3:98_1_1"/><i><font color="Green">E86</font></i>: 
 <a href="arytm_3.html#K14" target="_self">one</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T66" target="_self">Th66</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/><a NAME="E4:98_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#R3" target="_self">&lt;='</a>  <a href="arytm_3.html#K14" target="_self">one</a> 
 <b>by </b><i><a class="txt" href="arytm_3.html#E3:98_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="arytm_3.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E5:98_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="arytm_3.html#E3:98"><i><font color="Green">E84</font></i></a>, <a class="txt" href="arytm_3.html#E3:98_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="arytm_3.html#T64" target="_self">Th64</a></i>;<br/><a NAME="E6:98_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14" target="_self">one</a> 
 <b>by </b><i><a class="txt" href="arytm_3.html#E4:98_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="arytm_3.html#T73" target="_self">Th73</a></i>;<br/><b>hence </b><a NAME="E7:98_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:98_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="arytm_3.html#T74" target="_self">Th74</a></i>;<br/><b>assume </b><a NAME="E8:98_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><a NAME="E9:98_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
(  <a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="arytm_3.html#K9" target="_self">denominator</a> <a href="arytm_3.html#K14" target="_self">one</a>  <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="arytm_3.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E10:98_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K9" target="_self">denominator</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="arytm_3.html#K3" target="_self">*^</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/><b>then </b><i><font color="Green">E89</font></i>: 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_3.html#K2" target="_self">+^</a> <span class="p2">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="arytm_3.html#K10" target="_self">/</a> 1
<b>by </b><i><a class="txt" href="arytm_3.html#E3:98_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="arytm_3.html#E9:98_1_1"><i><font color="Green">E88</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_3.html#K2" target="_self">+^</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_3.html#T46" target="_self">Th46</a></i>
;<br/><a NAME="E12:98_1_1"/><b>then </b>
 <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> 1
 <b>by </b><i><a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/><a NAME="E13:98_1_1"/><b>then </b>
(  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  or  <a href="arytm_3.html#K8" target="_self">numerator</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="ordinal3.html#T19">ORDINAL3:19</a></i>;<br/><b>hence </b><a NAME="E14:98_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:98"><i><font color="Green">E83</font></i></a>, <a class="txt" href="arytm_3.html#E11:98_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_2"><b>suppose </b></a><a NAME="E1:98_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14" target="_self">one</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>4</sub></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><br/><a NAME="E3:98_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T66" target="_self">Th66</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/><a NAME="E4:98_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E3:98_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="arytm_3.html#D13" target="_self">Def13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:98_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:98_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E3:98_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="arytm_3.html#T56" target="_self">Th56</a></i>;<br/><b>hence </b><a NAME="E3:98_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:98"><i><font color="Green">E82</font></i></a>, <a class="ref" href="arytm_3.html#T69" target="_self">Th69</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:98_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E4:98_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="arytm_3.html#T73" target="_self">Th73</a></i>;<br/><a NAME="E7:98_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K14" target="_self">one</a> 
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:98_1_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="arytm_3.html#E4:98_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="arytm_3.html#T74" target="_self">Th74</a></i>;<br/><a NAME="E8:98_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  &amp;  <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:98"><i><font color="Green">E82</font></i></a>, <a class="txt" href="arytm_3.html#E3:98_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="arytm_3.html#T56" target="_self">Th56</a>, <a class="ref" href="arytm_3.html#T71" target="_self">Th71</a></i>;<br/><a NAME="E9:98_1_2"/><b>then </b>
(  <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; 1 <a href="hidden.html#R1">=</a> <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="arytm_3.html#K11" target="_self">+</a> <a href="arytm_3.html#K14" target="_self">one</a>  )
 <b>by </b><i><a class="ref" href="arytm_3.html#T56" target="_self">Th56</a>, <a class="ref" href="arytm_3.html#T73" target="_self">Th73</a></i>;<br/><b>hence </b><a NAME="E10:98_1_2"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="txt" href="arytm_3.html#E7:98_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="arytm_3.html#T79" target="_self">Th79</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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