<?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>set </b><font color="Maroon">AV</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : verum </span>}</span> ;<br/>
<a NAME="E1:113"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> iff for <font color="Olive">V</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>  holds <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">V</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><b>proof </b></a><div class="add">

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:113_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 ;<br/><b>let </b><font color="Maroon">V</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>;<br/><a NAME="E2:113_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><a NAME="E3:113_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:113_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/><b>hence </b><a NAME="E5:113_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T39">TOPS_1:39</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:113_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">V</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>  holds <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">V</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:113_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:113_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E3:113_1_2"/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E13"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="rltopsp1.html#E5"><i><font color="Green">Lemma45</font></i></a></i>;<br/><a NAME="E4:113_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/><a NAME="E5:113_1_2"/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T11" target="_self">Th11</a>, </i>;<br/><a NAME="E6:113_1_2"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/><a NAME="E7:113_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><a NAME="E8:113_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:113_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><a NAME="E10:113_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/><b>hence </b><a NAME="E11:113_1_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:113_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T39">TOPS_1:39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:113"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">V</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> holds <br/> ( <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">V</font></span>)</span> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">V</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:113_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 ;<br/><a NAME="E2:113_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K2" target="_self">-</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E13"><i><font color="Green">Lemma73</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:113_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span>
 ;<br/></div>
<b>end;</b></div>

<a NAME="E2:113_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">a</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">v</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>assume </b><a NAME="E3:113_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:113_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E6:113_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E7:113_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E13"><i><font color="Green">Lemma73</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">v'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:113_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v'</font>
 <b>and </b><br/><a NAME="E10:113_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">v'</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v'</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">v'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<b>then </b><i><font color="Green">E67</font></i>: <font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v'</font> = 
<font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rltopsp1.html#T11" target="_self">Th11</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<a NAME="E13:113_2"/>
<font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D9">RUSUB_4:def 9</a></i>;<br/>
<a NAME="E14:113_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:113_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">V</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>;<br/>
<a NAME="E4:113"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : verum </span>}</span> 
 
;<br/>
<b>thus </b><a NAME="E5:113"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:113_3"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:113_3_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">V</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:113_3_1"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E4:113_3_1"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">a</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">v</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font></span>)</span></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/><a NAME="E5:113_3_1"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">a</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">v</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/><a NAME="E6:113_3_1"/><i><font color="Green">E174</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E7:113_3_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T10" target="_self">Th10</a>, <a class="ref" href="rltopsp1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E8:113_3_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E10:113_3_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E11:113_3_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E12:113_3_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">v'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E14:113_3_1"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v'</font>
 <b>and </b><br/><a NAME="E15:113_3_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">v'</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E16:113_3_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><a NAME="E17:113_3_1"/><b>then </b>
<font color="Maroon">v'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">v''</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E19:113_3_1"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">v'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v''</font>
 <b>and </b><br/><a NAME="E20:113_3_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">v''</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v''</font>
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">v''</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/><b>hence </b><a NAME="E22:113_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a>, <a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:113_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="rltopsp1.html#E13"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E6:113"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E7:113"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:113_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:113_4"/><b>then </b>
 ex <font color="Olive">V</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font>
 
;<br/>
<b>hence </b><a NAME="E3:113_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">AV'</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : verum </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E9:113"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">AV'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="rltopsp1.html#T11" target="_self">Th11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">V</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>;<br/><a NAME="E1:113_5"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><a NAME="E2:113_5"/><b>then </b><i><font color="Green">E186</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/><a NAME="E3:113_5"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:113_5"/><b>then </b>
<font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/><a NAME="E5:113_5"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T11" target="_self">Th11</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:113_5"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:113"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T10" target="_self">Th10</a></i>;<br/>


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