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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_2.html#K2" target="_self">REAL+</a> ;<br/>

<b>assume </b><a NAME="E1:69"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/>

<a NAME="E2:69"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</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_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="ref" href="arytm_2.html#D5" target="_self">Def5</a>, <a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/>
<b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:69_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><b>hence </b><a NAME="E2:69_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a></i>;<br/><b>thus </b><a NAME="E5:69_1"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a></i>;<br/><a NAME="E6:69_1"/>
<font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:69_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T30">ORDINAL3:30</a></i>;<br/><a NAME="E9:69_1"/>
<font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</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="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/><a NAME="E11:69_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">C</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/><a NAME="E13:69_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">z'</font>
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a></i>;<br/><a NAME="E14:69_1"/><b>then </b>
<font color="Maroon">y'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/><b>hence </b><a NAME="E15:69_1"/>
<font color="Maroon">y</font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:69"><i><font color="Green">E19</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E6:69"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a>  ;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, <a class="ref" href="arytm_3.html#T34">ARYTM_3:34</a></i>;<br/>
<b>assume </b><a NAME="E9:69"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/>

<b>assume </b><a NAME="E10:69"/>
<font color="Maroon">y</font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E11:69"/><b>then </b>
<font color="Maroon">y'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:69"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">z'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E13:69"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">y'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</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_2.html#D5" target="_self">Def5</a>, <a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, , <a class="ref" href="arytm_3.html#T78">ARYTM_3:78</a></i>;<br/>
<a NAME="E15:69"/>
<font color="Maroon">n</font> <a href="arytm_3.html#K2">+^</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="txt" href="arytm_2.html#E22"><i><font color="Green">Lemma58</font></i></a></i>;<br/>
<a NAME="E16:69"/><b>then </b>
<font color="Maroon">n</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/>
<a NAME="E17:69"/><b>then </b>
<font color="Maroon">n</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E18:69"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>


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