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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">X2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>





<b>thus </b><a NAME="E1:142"/>
( <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R3">are_separated</a>  implies ( <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font> &amp; ( for <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a><br/> for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Olive">Y</font>  st <font color="Olive">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<font color="Olive">Y</font> &amp; <font color="Olive">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<font color="Olive">Y</font> holds <br/><font color="Olive">g</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Olive">Y</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:142_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R3">are_separated</a> 
 ;<br/>

<a NAME="E2:142_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4">are_weakly_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T85">TSEP_1:85</a></i>;<br/>
<b>thus </b><a NAME="E3:142_1"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tsep_1.html#T68">TSEP_1:68</a></i>;<br/>

<b>thus </b><a NAME="E4:142_1"/>
for <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a><br/> for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Olive">Y</font>  st <font color="Olive">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<font color="Olive">Y</font> &amp; <font color="Olive">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<font color="Olive">Y</font> holds <br/><font color="Olive">g</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Olive">Y</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, </i>;<br/>


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

<b>thus </b><a NAME="E2:142"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font> &amp; ( for <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a><br/> for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Olive">Y</font>  st <font color="Olive">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<font color="Olive">Y</font> &amp; <font color="Olive">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<font color="Olive">Y</font> holds <br/><font color="Olive">g</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Olive">Y</font> ) implies <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R3">are_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:142_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font>
 ;<br/>

<b>assume </b><a NAME="E2:142_2"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a><br/> for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Olive">Y</font>  st <font color="Olive">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<font color="Olive">Y</font> &amp; <font color="Olive">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<font color="Olive">Y</font> holds <br/><font color="Olive">g</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Olive">Y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">A1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E5:142_2"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#NR7" target="_self">are_not_separated</a> 
 ;<br/>

<a NAME="E6:142_2"/><b>then </b><i><font color="Green">E38</font></i>: 
 ex <font color="Olive">A10</font>, <font color="Olive">A20</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">A10</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> &amp; <font color="Olive">A20</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font> &amp; <font color="Olive">A10</font>,<font color="Olive">A20</font> <a href="tsep_1.html#NR3" target="_self">are_not_separated</a>  )
 
<b>by </b><i><a class="ref" href="tsep_1.html#D8">TSEP_1:def 8</a></i>;<br/>
<a NAME="E7:142_2"/><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#D2">TSEP_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">C1</font> = <font color="Maroon">A1</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C2</font> = <font color="Maroon">A2</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <b>by </b><i><a class="ref" href="tmap_1.html#T31" target="_self">Th31</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:142_2"/><i><font color="Green">E42</font></i>: 
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T47">PRE_TOPC:47</a></i>;<br/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="142_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:142_2_1"/>
<font color="Maroon">C1</font>,<font color="Maroon">C2</font> <a href="connsp_1.html#R1">are_separated</a> 
 ;<br/><a NAME="E2:142_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, <a class="ref" href="connsp_1.html#D1">CONNSP_1:def 1</a></i>;<br/><a NAME="E3:142_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:142_2_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/><a NAME="E5:142_2_1"/>
( <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> &amp; <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> )
 ;<br/><a NAME="E6:142_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> &amp; <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:142_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:142_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E9:142_2_1"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E10:142_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tmap_1.html#T29" target="_self">Th29</a>, <a class="ref" href="connsp_1.html#D1">CONNSP_1:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:142_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">C1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C2</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y1</font> = <font color="Maroon">X1</font>, <font color="Maroon">Y2</font> = <font color="Maroon">X2</font> as    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> <b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="142_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:142_2_2_1"/>
(  not <font color="Maroon">C1</font> is <a href="pre_topc.html#V3">open</a> or  not <font color="Maroon">C2</font> is <a href="pre_topc.html#V3">open</a> )
 </span><b>by </b><i><a class="ref" href="tmap_1.html#T33" target="_self">Th33</a>, <a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tsep_1.html#T41">TSEP_1:41</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_2_2_1_1"><b>suppose </b></a><a NAME="E1:142_2_2_1_1"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">C1</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><div class="add"><b>set </b><font color="Maroon">Y</font> = <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">C1</font>;<br/><b>set </b><font color="Maroon">g</font> =  <a href="tmap_1.html#K7" target="_self">modid</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C1</font>;<br/><a NAME="E2:142_2_2_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="tmap_1.html#K7" target="_self">modid</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C1</font> is  not  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">C1</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, </i>;<br/><a NAME="E3:142_2_2_1_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C1</font></span>)</span> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">Y1</font> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C1</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font>
 <b>by </b><i/>;<br/><a NAME="E4:142_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C1</font></span>)</span> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">Y2</font> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C1</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font>
 <b>by </b><i/>;<br/><a NAME="E5:142_2_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C1</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">C1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C1</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">C1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T37" target="_self">Th37</a>, <a class="ref" href="tmap_1.html#T17" target="_self">Th17</a>, </i>;<br/><b>hence </b><a NAME="E6:142_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_2_2_1_2"><b>suppose </b></a><a NAME="E1:142_2_2_1_2"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">C2</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><div class="add"><b>set </b><font color="Maroon">Y</font> = <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">C2</font>;<br/><b>set </b><font color="Maroon">g</font> =  <a href="tmap_1.html#K7" target="_self">modid</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C2</font>;<br/><a NAME="E2:142_2_2_1_2"/><i><font color="Green">E50</font></i>: 
 <a href="tmap_1.html#K7" target="_self">modid</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C2</font> is  not  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">C2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, </i>;<br/><a NAME="E3:142_2_2_1_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C2</font></span>)</span> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">Y1</font> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font>
 <b>by </b><i/>;<br/><a NAME="E4:142_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C2</font></span>)</span> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">Y2</font> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font>
 <b>by </b><i/>;<br/><a NAME="E5:142_2_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">C2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">C2</font></span>)</span> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">C2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T37" target="_self">Th37</a>, <a class="ref" href="tmap_1.html#T17" target="_self">Th17</a>, </i>;<br/><b>hence </b><a NAME="E6:142_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E15:142_2"/>
contradiction
 ;<br/>


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


</div>
