<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="compts_1.html#V5" title="COMPTS_1:attr.5">being_T4</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">G</font> be    <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>;<br/><b>let </b><font color="Maroon" title="c5">r1</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> ;<br/>



<b>assume </b><a NAME="E2:30"/><i><font color="Green" title="E28">A2</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r1</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:30"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="urysohn3.html#K7" title="URYSOHN3:func.7">.</a> <font color="Maroon" title="c6">p</font>
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c5">r1</font>
 ;<br/>

<a NAME="E5:30"/>
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12" title="PROB_1:func.12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 
<b>by </b><i><a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:30"/><b>then </b>
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">halfline</a> 0 or <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:30"/><b>then </b>
<font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E2:30"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="urysohn1.html#T1" title="URYSOHN1:th.1">URYSOHN1:1</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:30"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:30"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="urysohn3.html#E2:30"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="urysohn3.html#E3:30"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="urysohn3.html#E4:30"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="urysohn3.html#T19" target="_self" title="URYSOHN3:th.19">Th19</a></i>;<br/>


</div>
