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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:96"/>
( <font color="Maroon">T</font> is <a href="compts_1.html#V3">Hausdorff</a> implies for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Olive">N</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Olive">N</font> holds <br/><font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:96_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">T</font> is <a href="compts_1.html#V3">Hausdorff</a>
 ;<br/>

<b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/>

<b>given </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><a NAME="E2:96_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E3:96_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E4:96_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 ;<br/>

<b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:96_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E7:96_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E8:96_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E9:96_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E10:96_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T10" target="_self">Th10</a>, <a class="txt" href="yellow_6.html#E16"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a></i>;<br/>
<a NAME="E11:96_1"/>
<font color="Maroon">W</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p1</font>
 
<b>by </b><i>, , <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<a NAME="E12:96_1"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:96_1"/>
<font color="Maroon">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p2</font>
 
<b>by </b><i>, <a class="txt" href="yellow_6.html#E17"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<a NAME="E14:96_1"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D5" target="_self">Def5</a>, </i>;<br/>
<b>hence </b><a NAME="E15:96_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


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

<b>assume </b><a NAME="E2:96"/><i><font color="Green">E128</font></i>: 
for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Olive">N</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Olive">N</font> holds <br/><font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font>
 ;<br/>

<b>given </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><a NAME="E3:96"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E4:96"/><i><font color="Green">E147</font></i>: 
for <font color="Olive">W</font>, <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> holds <br/><font color="Olive">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">V</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a><br/>

<b>set </b><font color="Maroon">pN</font> = <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>;<br/>
<b>set </b><font color="Maroon">cT</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">cpN</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>;<br/>
<a NAME="E5:96"/><i><font color="Green">E149</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="yellow_3.html#K9">`2</a> </span>)</span>;<br/>
<a NAME="E6:96"/><i><font color="Green">E181</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>  holds the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:96_2"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="yellow_3.html#K8">`1</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:96_2"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="yellow_3.html#K9">`2</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>
<a NAME="E5:96_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
;<br/>
<a NAME="E6:96_2"/>
<font color="Maroon">i</font> <a href="yellow_3.html#K8">`1</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">i</font> <a href="yellow_3.html#K9">`2</a> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#D5" target="_self">Def5</a>, , </i>;<br/>
<a NAME="E7:96_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E8:96_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="yellow_3.html#K9">`2</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="txt" href="yellow_6.html#E17"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E9:96_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="yellow_3.html#K9">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:96"/><i><font color="Green">E185</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="nattra_1.html#S1">NATTRA_1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">N</font> =  <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>,<font color="Maroon">f</font> #) as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_3"><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 <font color="Maroon">p</font>;<br/><a NAME="E1:96_3"/><i><font color="Green">E186</font></i>: 
 <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="E2:96_3"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:96_3_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a>, <a class="ref" href="tops_1.html#T51">TOPS_1:51</a></i>;<br/>
<a NAME="E2:96_3_1"/><b>then </b><i><font color="Green">E187</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:96_3_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T53">TOPS_1:53</a></i>;<br/>
<a NAME="E4:96_3_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">i</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K4">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i><a class="txt" href="yellow_6.html#E16"><i><font color="Green">Lemma60</font></i></a>, <a class="txt" href="yellow_6.html#E17"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>take </b>
<font color="Maroon">i</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>

<b>consider </b><font color="Maroon">j1</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>, <font color="Maroon">j2</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span><b> such that </b><br/><a NAME="E7:96_3_1"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">j1</font>,<font color="Maroon">j2</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E16"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E8:96_3_1"/><i><font color="Green">E191</font></i>: 
( <font color="Maroon">j</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>consider </b><font color="Maroon">W1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E10:96_3_1"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">W2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E12:96_3_1"/><i><font color="Green">E195</font></i>: 
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">W2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font>, <font color="Maroon">i'</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> ;<br/>
<a NAME="E14:96_3_1"/><i><font color="Green">E196</font></i>: 
( <font color="Maroon">i'</font> <a href="yellow_3.html#K8">`1</a>  <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> &amp; <font color="Maroon">i'</font> <a href="yellow_3.html#K9">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>assume </b><a NAME="E15:96_3_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E16:96_3_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E17:96_3_1"/><b>then </b>
<font color="Maroon">i'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j'</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E18:96_3_1"/><b>then </b>
( <font color="Maroon">i'</font> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j'</font> <a href="yellow_3.html#K8">`1</a>  &amp; <font color="Maroon">i'</font> <a href="yellow_3.html#K9">`2</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j'</font> <a href="yellow_3.html#K9">`2</a>  )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T12">YELLOW_3:12</a></i>;<br/>
<a NAME="E19:96_3_1"/><b>then </b><i><font color="Green">E197</font></i>: 
( <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> &amp; <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#T11" target="_self">Th11</a>, , <a class="ref" href="yellow_6.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E20:96_3_1"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_6.html#T11" target="_self">Th11</a>, </i>;<br/>
<a NAME="E21:96_3_1"/>
<font color="Maroon">W1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E22:96_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E23:96_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E24:96_3_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:96_3"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T8">WAYBEL_0:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:96"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_4"><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 <font color="Maroon">q</font>;<br/><a NAME="E1:96_4"/><i><font color="Green">E200</font></i>: 
 <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="E2:96_4"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:96_4_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a>, <a class="ref" href="tops_1.html#T51">TOPS_1:51</a></i>;<br/>
<a NAME="E2:96_4_1"/><b>then </b><i><font color="Green">E201</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:96_4_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T53">TOPS_1:53</a></i>;<br/>
<a NAME="E4:96_4_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">i</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span></span><a href="tarski.html#K4">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i><a class="txt" href="yellow_6.html#E16"><i><font color="Green">Lemma60</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>take </b>
<font color="Maroon">i</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>

<b>consider </b><font color="Maroon">j1</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>, <font color="Maroon">j2</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span><b> such that </b><br/><a NAME="E7:96_4_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">j1</font>,<font color="Maroon">j2</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E16"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E8:96_4_1"/><i><font color="Green">E203</font></i>: 
( <font color="Maroon">j</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>consider </b><font color="Maroon">W1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E10:96_4_1"/><i><font color="Green">E204</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> &amp; <font color="Maroon">W1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">W2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E12:96_4_1"/><i><font color="Green">E205</font></i>: 
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W2</font> &amp; <font color="Maroon">W2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font>, <font color="Maroon">i'</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> ;<br/>
<a NAME="E14:96_4_1"/><i><font color="Green">E206</font></i>: 
( <font color="Maroon">i'</font> <a href="yellow_3.html#K8">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp; <font color="Maroon">i'</font> <a href="yellow_3.html#K9">`2</a>  <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>assume </b><a NAME="E15:96_4_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E16:96_4_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" target="_self">OpenNeighborhoods</a> <font color="Maroon">q</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E17:96_4_1"/><b>then </b>
<font color="Maroon">i'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j'</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E18:96_4_1"/><b>then </b>
( <font color="Maroon">i'</font> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j'</font> <a href="yellow_3.html#K8">`1</a>  &amp; <font color="Maroon">i'</font> <a href="yellow_3.html#K9">`2</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j'</font> <a href="yellow_3.html#K9">`2</a>  )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T12">YELLOW_3:12</a></i>;<br/>
<a NAME="E19:96_4_1"/><b>then </b><i><font color="Green">E207</font></i>: 
( <font color="Maroon">W1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp; <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a>, , , <a class="ref" href="yellow_6.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E20:96_4_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W2</font>
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#T11" target="_self">Th11</a>, , </i>;<br/>
<a NAME="E21:96_4_1"/>
<font color="Maroon">W1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E22:96_4_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E23:96_4_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E24:96_4_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:96_4"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E17"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="waybel_0.html#T8">WAYBEL_0:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:96"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:96"/>
contradiction
 <b>by </b><i><a class="ref" href="yellow_6.html#T10" target="_self">Th10</a>, , </i>;<br/>


</div>
