<?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="E1:143"/>
( <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">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Olive">Y</font>  st <font color="Olive">f</font> <a href="tmap_1.html#K2" 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">f</font> <a href="tmap_1.html#K2" 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">f</font> <a href="tmap_1.html#K2" target="_self">|</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> 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="143_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:143_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="E2:143_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="E3:143_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#T27" target="_self">Th27</a>, <a class="ref" href="tsep_1.html#T68">TSEP_1:68</a></i>;<br/>

<b>thus </b><a NAME="E4:143_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">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Olive">Y</font>  st <font color="Olive">f</font> <a href="tmap_1.html#K2" 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">f</font> <a href="tmap_1.html#K2" 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">f</font> <a href="tmap_1.html#K2" target="_self">|</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> 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#T29" target="_self">Th29</a>, <a class="ref" href="tmap_1.html#T20" target="_self">Th20</a></i>;<br/>


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

<b>thus </b><a NAME="E2:143"/>
( <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">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Olive">Y</font>  st <font color="Olive">f</font> <a href="tmap_1.html#K2" 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">f</font> <a href="tmap_1.html#K2" 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">f</font> <a href="tmap_1.html#K2" target="_self">|</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> 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="143_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:143_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="E2:143_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">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Olive">Y</font>  st <font color="Olive">f</font> <a href="tmap_1.html#K2" 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">f</font> <a href="tmap_1.html#K2" 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">f</font> <a href="tmap_1.html#K2" target="_self">|</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> 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/>

<a NAME="E3:143_2"/>
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="143_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</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">g</font> be   <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="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:143_2_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">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="Maroon">Y</font> &amp; <font color="Maroon">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="Maroon">Y</font> )
 ;<br/>

<b>consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:143_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">h</font> <a href="tmap_1.html#K2" target="_self">|</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> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:143_2_1"/>
( <font color="Maroon">X1</font> is    <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> &amp; <font color="Maroon">X2</font> is    <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/>
<a NAME="E5:143_2_1"/><b>then </b>
( <font color="Maroon">h</font> <a href="tmap_1.html#K2" 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="Maroon">Y</font> &amp; <font color="Maroon">h</font> <a href="tmap_1.html#K2" 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="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T31" target="_self">Th31</a>, <a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, </i>;<br/>
<b>hence </b><a NAME="E6:143_2_1"/>
<font color="Maroon">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="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T29" target="_self">Th29</a>, <a class="ref" href="tmap_1.html#T32" target="_self">Th32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:143_2"/>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R3">are_separated</a> 
 <b>by </b><i><a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a></i>;<br/>


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


</div>
