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<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> ;<br/><b>let </b><font color="Maroon">i</font> be  <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>assume </b><a NAME="E1:97"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">i</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="arytm_3.html#K11" target="_self">+</a> <a href="arytm_3.html#K14" target="_self">one</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 ;<br/>

<a NAME="E2:97"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</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="E4:97"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T30" target="_self">Th30</a>, <a class="ref" href="arytm_3.html#T23" target="_self">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon">z</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="E6:97"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">i</font> <a href="arytm_3.html#K11" target="_self">+</a> <a href="arytm_3.html#K14" target="_self">one</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T30" target="_self">Th30</a>, <a class="ref" href="arytm_3.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E7:97"/>
<font color="Maroon">i</font> <a href="arytm_3.html#K11" target="_self">+</a> <a href="arytm_3.html#K14" target="_self">one</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="arytm_3.html#K2" target="_self">+^</a> 1
 
<b>by </b><i><a class="ref" href="arytm_3.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E8:97"/><b>then </b>
<font color="Maroon">i</font> <a href="arytm_3.html#K11" target="_self">+</a> <a href="arytm_3.html#K14" target="_self">one</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  <b>by </b><i><a class="ref" href="arytm_3.html#T30" target="_self">Th30</a>, <a class="ref" href="arytm_3.html#T31" target="_self">Th31</a>, <a class="ref" href="arytm_3.html#T32" target="_self">Th32</a>, <a class="ref" href="arytm_3.html#T33" target="_self">Th33</a>, <a class="ref" href="arytm_3.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E10:97"/>
<font color="Maroon">i</font> <a href="arytm_3.html#K11" target="_self">+</a> <a href="arytm_3.html#K14" target="_self">one</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="arytm_3.html#K11" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T32" target="_self">Th32</a>, <a class="ref" href="arytm_3.html#T33" target="_self">Th33</a>, <a class="ref" href="arytm_3.html#T17" target="_self">Th17</a></i>;<br/>
<b>then </b>1 = 
<font color="Maroon">y</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T24" target="_self">Th24</a></i>
<br/>.= 
<font color="Maroon">y'</font> <a href="arytm_3.html#K2" target="_self">+^</a> <font color="Maroon">z'</font>
<b>by </b><i><a class="ref" href="arytm_3.html#T21" target="_self">Th21</a></i>
;<br/>
<a NAME="E12:97"/><b>then </b>
( <font color="Maroon">y'</font> <a href="ordinal1.html#R1">c=</a> 1 &amp; <font color="Maroon">z'</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="E13:97"/><b>then </b>
( ( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i><a class="ref" href="ordinal3.html#T19">ORDINAL3:19</a></i>;<br/>
<a NAME="E14:97"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">i</font> <a href="arytm_3.html#K11" target="_self">+</a> <a href="arytm_3.html#K14" target="_self">one</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T32" target="_self">Th32</a>, </i>;<br/>
<b>hence </b><a NAME="E15:97"/>
contradiction
 <b>by </b><i><a class="ref" href="arytm_3.html#T30" target="_self">Th30</a></i>;<br/>


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