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<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">RAT+</a> , <font color="Maroon">j</font> be  <a href="ordinal1.html#V3">ordinal</a>  <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/>

<a NAME="E1:13"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T37">ARYTM_3:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">i</font>, <font color="Maroon">y</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  <b>by </b><i><a class="ref" href="arytm_2.html#T2">ARYTM_2:2</a></i>;<br/>
<b>assume </b><a NAME="E3:13"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E4:13"/><b>then </b>
<font color="Maroon">x</font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="arytm_2.html#T19">ARYTM_2:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">y'</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="E6:13"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> )
 <b>and </b><br/><a NAME="E7:13"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#D5">ARYTM_2:def 5</a></i>;<br/>
<a NAME="E8:13"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="numbers.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:13"/>
<font color="Maroon">i</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="numbers.html#E3"><i><font color="Green">Lemma12</font></i></a>, <a class="txt" href="numbers.html#E1:13"><i><font color="Green">E13</font></i></a>, <a class="ref" href="arytm_3.html#T73">ARYTM_3:73</a></i>;<br/>


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