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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V5">being_T4</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">B</font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a>  &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/>

<b>let </b><font color="Maroon">G</font> be    <a href="urysohn3.html#M2" target="_self">Rain</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">r1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K3">DOM</a> ;<br/>



<b>assume </b><a NAME="E2:31"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:31"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E4:31"/><i><font color="Green">E36</font></i>: 
<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">G</font></span>)</span> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">r1</font>
 ;<br/>

<a NAME="E5:31"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 )
 
<b>by </b><i><a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:31"/><b>then </b>
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 or <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:31"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <a href="urysohn1.html#K2">DYADIC</a>  <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="urysohn1.html#T1">URYSOHN1:1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:31"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/>


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