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<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7" target="_self">RAT+</a> ;<br/>

<b>assume </b><a NAME="E1:115"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7" target="_self">RAT+</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> <b> such that </b><a NAME="E1:115_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a>  &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 ;<br/><a NAME="E2:115_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:115_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:115_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><a NAME="E2:115_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E3:115_1_1"/><b>then </b>
 <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/><a NAME="E4:115_1_1"/><b>then </b>
 <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:115_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="arytm_3.html#E1"><i><font color="Green">Lemma9</font></i></a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> <b> such that </b><br/><a NAME="E6:115_1"/><i><font color="Green">E16</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">i1</font>,<font color="Maroon">j1</font> <a href="arytm_3.html#R1" target="_self">are_relative_prime</a>  &amp; <font color="Maroon">j1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  &amp; <font color="Maroon">j1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <b>by </b><i><a class="ref" href="arytm_3.html#T34" target="_self">Th34</a>, </i>;<br/><a NAME="E7:115_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i1</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K2">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E8:115_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i1</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E9:115_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i1</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E10:115_1"/><b>then </b>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E11:115_1"/><b>then </b>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="arytm_3.html#K3" target="_self">*^</a> 1 )
 <b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E12:115_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E4:115"/><i><font color="Green">E17</font></i>: 
 <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:115_2"/>
<font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/><a NAME="E2:115_2"/><b>then </b>
 <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a></i>;<br/><b>hence </b><a NAME="E3:115_2"/>
contradiction
 <b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:115"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>
<a NAME="E8:115"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T34" target="_self">Th34</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as  <a href="ordinal1.html#V3">ordinal</a>  <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a>  ;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E10:115"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T34" target="_self">Th34</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>

<b>let </b><font color="Maroon">r</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="E11:115"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E12:115"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as  <a href="ordinal1.html#V3">ordinal</a>  <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a>  <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E14:115"/>
<font color="Maroon">C</font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">C</font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">r</font> as  <a href="ordinal1.html#V3">ordinal</a>  <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a>  <b>by </b><i/>;<br/>
<a NAME="E16:115"/>
<font color="Maroon">r</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T34" target="_self">Th34</a>, , <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<b>then </b><font color="Maroon">C</font> = 
<font color="Maroon">r</font> <a href="arytm_3.html#K2" target="_self">+^</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="ordinal3.html#D6">ORDINAL3:def 6</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T21" target="_self">Th21</a></i>
;<br/>
<b>hence </b><a NAME="E18:115"/>
<font color="Maroon">r</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T23" target="_self">Th23</a></i>;<br/>


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