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

<b>let </b><font color="Maroon">X</font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:103"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> <a href="urysohn1.html#NR1" target="_self">is_T1</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:103_1"/>
( <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1"><b>suppose </b></a><a NAME="E1:103_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:103_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><b>hence </b>0 <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> = 
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="convex1.html#T33">CONVEX1:33</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default">0 <a href="convex1.html#K1">*</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="convex1.html#T33">CONVEX1:33</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_2"><b>suppose </b></a><a NAME="E1:103_1_2"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:103_1_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E3:103_1_2"/><b>then </b>
not  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b>0 <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="convex1.html#T34">CONVEX1:34</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, <a class="ref" href="yellow_8.html#T35">YELLOW_8:35</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default">0 <a href="convex1.html#K1">*</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i>, <a class="ref" href="convex1.html#T34">CONVEX1:34</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
