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

<b>assume </b><a NAME="E1:112_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> is <a href="urysohn1.html#V1">being_T1</a>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">q</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="compts_1.html#D4">COMPTS_1:def 4</a><br/>

<b>assume </b><a NAME="E2:112_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E3:112_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T17">ZFMISC_1:17</a></i>;<br/>
<a NAME="E4:112_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T41">BORSUK_1:41</a></i>;<br/>
<a NAME="E5:112_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T10" target="_self">Th10</a>, <a class="ref" href="urysohn1.html#T27">URYSOHN1:27</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="E7:112_1_1"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="txt" href="rltopsp1.html#E13"><i><font color="Green">Lemma73</font></i></a>, </i>;<br/>
<a NAME="E8:112_1_1"/><i><font color="Green">E49</font></i>: 
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q</font></span><a href="domain_1.html#K6">}</a></span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="rusub_4.html#T36">RUSUB_4:36</a></i>;<br/>
<b>take </b>
<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>
;<br/>

<b>take </b>
<font color="Maroon">q</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/>

<b>thus </b><a NAME="E9:112_1_1"/>
( <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> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">q</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> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<a NAME="E10:112_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="E11:112_1_1"/><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> &amp; <font color="Maroon">q</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">q</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="E12:112_1_1"/>
( <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> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</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/>

<a NAME="E13:112_1_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</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="E14:112_1_1"/><b>then </b>
( <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> <font color="Maroon">p</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font> &amp; <font color="Maroon">q</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> <font color="Maroon">q</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:112_1_1"/>
<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="xboole_0.html#R1">misses</a> <font color="Maroon">q</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#T11" target="_self">Th11</a>, , <a class="ref" href="xboole_1.html#T64">XBOOLE_1:64</a></i>;<br/>


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