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<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> 
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green">E24</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">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E1:23"><i><font color="Green">E23</font></i></a>, <a class="txt" href="arytm_2.html#E2:23"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></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="E5:23"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E6:23"/>
 <a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/>
<b>hence </b><a NAME="E7:23"/>
 <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:23"><i><font color="Green">E23</font></i></a>, <a class="txt" href="arytm_2.html#E5:23"><i><font color="Green">E25</font></i></a>, <a class="txt" href="arytm_2.html#E26"><i><font color="Green">Lemma21</font></i></a></i>;<br/>


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