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<b>let </b><font color="Maroon">X</font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon">O</font> be    <a href="yellow13.html#M1">basis</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:91"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">P</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="rusub_4.html#K5">+</a> <font color="Olive">U</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">U</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> </span>}</span> 
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D4">YELLOW13:def 4</a><br/><b>let </b><font color="Maroon">A</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D2">YELLOW13:def 2</a><br/>



<a NAME="E2:91"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E3:91"/><i><font color="Green">E38</font></i>: 
 <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">p</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:91"/>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</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>, <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<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="E6:91"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>and </b><br/><a NAME="E7:91"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow13.html#D2">YELLOW13:def 2</a></i>;<br/>
<b>take </b><font color="Maroon">U</font> = <font color="Maroon">p</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font>;<br/>

<a NAME="E8:91"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">p</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="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">U</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E7:91"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E9:91"/>
 <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="E10:91"/><b>then </b>
<font color="Maroon">p</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">p</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="E11:91"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</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="E12:91"/>
<font color="Maroon">U</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a>, </i>;<br/>

<b>thus </b><a NAME="E13:91"/>
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#T5" target="_self">Th5</a></i>;<br/>

<a NAME="E14:91"/><i><font color="Green">E53</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E15:91"/>
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</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">p</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E16:91"/><b>then </b>
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E17:91"/><b>then </b>
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</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> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/>
<a NAME="E18:91"/><b>then </b>
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E19:91"/>
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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