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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="rltopsp1.html#V9" target="_self">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D12" target="_self">RLTOPSP1:def 12</a><br/>





<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:123"/><i><font color="Green">E77</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T60" target="_self">Th60</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:123"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:123"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D12" target="_self">Def12</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E6:123"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:123"><i><font color="Green">E78</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E7:123"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E8:123"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E5:123"><i><font color="Green">E79</font></i></a></i>;<br/>
<a NAME="E9:123"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E10:123"/><i><font color="Green">E82</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T53" target="_self">Th53</a>, <a class="txt" href="rltopsp1.html#E4:123"><i><font color="Green">E78</font></i></a>, <a class="txt" href="rltopsp1.html#E7:123"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E11:123"/>
<font color="Maroon">c<sub>6</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E2:123"><i><font color="Green">E77</font></i></a>, <a class="ref" href="convex1.html#T39">CONVEX1:39</a></i>;<br/>
<b>hence </b><a NAME="E12:123"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E9:123"><i><font color="Green">E81</font></i></a>, <a class="txt" href="rltopsp1.html#E10:123"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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