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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>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/>



<b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green">E19</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; ( for <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   holds <br/>(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> or  not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> or  not <font color="Olive">x'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">y'</font> ) ) ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> ) or ( not <!--HUMANRECFAILED-->( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) &amp; not <!--HUMANRECFAILED-->( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) &amp; not <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> ) )
 ;<br/>

<b>thus </b><a NAME="E2:18_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  implies  ex <font color="Olive">y'</font>, <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Olive">y'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">x'</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  )
 ;<br/>

<b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font>, <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>  ;<br/>
<a NAME="E3:18_1_1_1"/>
<font color="Maroon">y'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">x'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:18_1_1_1"/>
 ex <font color="Olive">y'</font>, <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Olive">y'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">x'</font> )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:18_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E4:18_1_1"/>
( not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  implies  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E5:18_1_1"/><i><font color="Green">E20</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) &amp; not <!--HUMANRECFAILED-->( not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) )
 ;<br/>

<a NAME="E6:18_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a>  )
 
;<br/>
<a NAME="E7:18_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) &amp;  ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) ) </span>}</span>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) &amp;  ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) ) </span>}</span>  )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:18_1_1"><i><font color="Green">E19</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:18_1_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E1:18_1_1"><i><font color="Green">E19</font></i></a>, </i>;<br/>


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