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<b>let </b><font color="Maroon">R1</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">R2</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; ( for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> holds <br/> not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> ) ) )
 <b>and </b><br/><a NAME="E2:22_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; ( for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> holds <br/> not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> ) ) )
 ;<br/>

<a NAME="E3:22_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font>
 ;<br/><a NAME="E2:22_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; ( for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> holds <br/> not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:22_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:22_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font>
 ;<br/>

<a NAME="E3:22_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; ( for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> holds <br/> not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:22_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_1_1"/>
<font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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