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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:96"/>
( <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V3" title="COMPTS_1:attr.3">Hausdorff</a> implies for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Olive" title="b1">N</font> holds <br/><font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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" title="E57">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V3" title="COMPTS_1:attr.3">Hausdorff</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>given </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><a NAME="E2:96_1"/><i><font color="Green" title="E58">A2</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c2">N</font>
 <b>and </b><br/><a NAME="E3:96_1"/><i><font color="Green" title="E59">A3</font></i>: 
<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c2">N</font>
 <b>and </b><br/><a NAME="E4:96_1"/><i><font color="Green" title="E60">A4</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">W</font>, <font color="Maroon" title="c6">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:96_1"/><i><font color="Green" title="E61">A5</font></i>: 
<font color="Maroon" title="c5">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E7:96_1"/><i><font color="Green" title="E62">A6</font></i>: 
<font color="Maroon" title="c6">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E8:96_1"/><i><font color="Green" title="E63">A7</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W</font>
 <b>and </b><br/><a NAME="E9:96_1"/><i><font color="Green" title="E64">A8</font></i>: 
<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font>
 <b>and </b><br/><a NAME="E10:96_1"/><i><font color="Green" title="E65">A9</font></i>: 
<font color="Maroon" title="c5">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">V</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:96_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="yellow_6.html#E4:96_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="compts_1.html#D4" title="COMPTS_1:def.4">COMPTS_1:def 4</a></i>;<br/>
<a NAME="E11:96_1"/>
<font color="Maroon" title="c5">W</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">p1</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E6:96_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="yellow_6.html#E8:96_1"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="connsp_2.html#T5" title="CONNSP_2:th.5">CONNSP_2:5</a></i>;<br/>
<a NAME="E12:96_1"/><b>then </b><i><font color="Green" title="E66">A10</font></i>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c5">W</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E2:96_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="yellow_6.html#D18" target="_self" title="YELLOW_6:def.18">Def18</a></i>;<br/>
<a NAME="E13:96_1"/>
<font color="Maroon" title="c6">V</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">p2</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:96_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="txt" href="yellow_6.html#E9:96_1"><i><font color="Green" title="E64">A8</font></i></a>, <a class="ref" href="connsp_2.html#T5" title="CONNSP_2:th.5">CONNSP_2:5</a></i>;<br/>
<a NAME="E14:96_1"/><b>then </b>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c6">V</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E3:96_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="yellow_6.html#D18" target="_self" title="YELLOW_6:def.18">Def18</a></i>;<br/>
<b>hence </b><a NAME="E15:96_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow_6.html#E10:96_1"><i><font color="Green" title="E65">A9</font></i></a>, <a class="txt" href="yellow_6.html#E12:96_1"><i><font color="Green" title="E66">A10</font></i></a>, <a class="ref" href="yellow_6.html#T26" target="_self" title="YELLOW_6:th.26">Th26</a></i>;<br/>


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

<b>assume </b><a NAME="E2:96"/><i><font color="Green" title="E57">A11</font></i>: 
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Olive" title="b1">N</font> holds <br/><font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><a NAME="E3:96"/><i><font color="Green" title="E58">A12</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <b>and </b><br/><a NAME="E4:96"/><i><font color="Green" title="E59">A13</font></i>: 
for <font color="Olive" title="b1">W</font>, <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> holds <br/><font color="Olive" title="b1">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">V</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D4" title="COMPTS_1:def.4">COMPTS_1:def 4</a><br/>

<b>set </b><font color="Maroon" title="c4">pN</font> = <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">cT</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/>
<b>set </b><font color="Maroon" title="c6">cpN</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/>
<a NAME="E5:96"/><i><font color="Green" title="E60">A14</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default">$1 <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default">$1 <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span>;<br/>
<a NAME="E6:96"/><i><font color="Green" title="E61">A15</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>  holds the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">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" title="c7">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/>

<b>consider </b><font color="Maroon" title="c8">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E2:96_2"/><i><font color="Green" title="E62">A16</font></i>: 
( <font color="Maroon" title="c8">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font> &amp; <font color="Maroon" title="c8">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T38" target="_self" title="YELLOW_6:th.38">Th38</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:96_2"/><i><font color="Green" title="E63">A17</font></i>: 
( <font color="Maroon" title="c9">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> &amp; <font color="Maroon" title="c9">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T38" target="_self" title="YELLOW_6:th.38">Th38</a></i>;<br/>
<a NAME="E5:96_2"/><i><font color="Green" title="E64">A18</font></i>: 
<font color="Maroon" title="c8">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
;<br/>
<a NAME="E6:96_2"/>
<font color="Maroon" title="c7">i</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c7">i</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> 
 
<b>by </b><i><a class="txt" href="yellow_6.html#E4:96"><i><font color="Green" title="E59">A13</font></i></a>, <a class="txt" href="yellow_6.html#E2:96_2"><i><font color="Green" title="E62">A16</font></i></a>, <a class="txt" href="yellow_6.html#E4:96_2"><i><font color="Green" title="E63">A17</font></i></a></i>;<br/>
<a NAME="E7:96_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E8:96_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="yellow_6.html#E2:96_2"><i><font color="Green" title="E62">A16</font></i></a>, <a class="txt" href="yellow_6.html#E4:96_2"><i><font color="Green" title="E63">A17</font></i></a>, <a class="txt" href="yellow_6.html#E5:96_2"><i><font color="Green" title="E64">A18</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E9:96_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E8:96"/><i><font color="Green" title="E62">A19</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> holds  <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S11" title="FUNCT_2:sch.11">FUNCT_2:sch 11</a>(<a class="txt" href="yellow_6.html#E6:96"><i><font color="Green" title="E61">A15</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c8">N</font> =  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>,<font color="Maroon" title="c7">f</font> #) as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="yellow_6.html#E17"><i><font color="Green" title="E9">Lm3</font></i></a>, <a class="txt" href="yellow_6.html#E18"><i><font color="Green" title="E10">Lm4</font></i></a></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" title="c9">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">p</font>;<br/><a NAME="E1:96_3"/><i><font color="Green" title="E63">A20</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">V</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/><a NAME="E2:96_3"/>
<font color="Maroon" title="c8">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">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" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a>, <a class="ref" href="tops_1.html#T51" title="TOPS_1:th.51">TOPS_1:51</a></i>;<br/>
<a NAME="E2:96_3_1"/><b>then </b><i><font color="Green" title="E64">A21</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T39" target="_self" title="YELLOW_6:th.39">Th39</a></i>;<br/>
<a NAME="E3:96_3_1"/>
( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T53" title="TOPS_1:th.53">TOPS_1:53</a></i>;<br/>
<a NAME="E4:96_3_1"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T39" target="_self" title="YELLOW_6:th.39">Th39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">i</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">N</font> <b>by </b><i><a class="txt" href="yellow_6.html#E5:96"><i><font color="Green" title="E60">A14</font></i></a>, <a class="txt" href="yellow_6.html#E2:96_3_1"><i><font color="Green" title="E64">A21</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">i</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon" title="c11">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">N</font>;<br/>

<b>consider </b><font color="Maroon" title="c12">j1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>, <font color="Maroon" title="c13">j2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span><b> such that </b><br/><a NAME="E7:96_3_1"/><i><font color="Green" title="E65">A22</font></i>: 
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c12">j1</font>,<font color="Maroon" title="c13">j2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E5:96"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E8:96_3_1"/><i><font color="Green" title="E66">A23</font></i>: 
( <font color="Maroon" title="c11">j</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j1</font> &amp; <font color="Maroon" title="c11">j</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:96_3_1"><i><font color="Green" title="E65">A22</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">W1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E10:96_3_1"/><i><font color="Green" title="E67">A24</font></i>: 
( <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">W1</font> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W1</font> &amp; <font color="Maroon" title="c14">W1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T38" target="_self" title="YELLOW_6:th.38">Th38</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">W2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E12:96_3_1"/><i><font color="Green" title="E68">A25</font></i>: 
( <font color="Maroon" title="c13">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">W2</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">W2</font> &amp; <font color="Maroon" title="c15">W2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T38" target="_self" title="YELLOW_6:th.38">Th38</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">j'</font> = <font color="Maroon" title="c11">j</font>, <font color="Maroon" title="c17">i'</font> = <font color="Maroon" title="c10">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> ;<br/>
<a NAME="E14:96_3_1"/><i><font color="Green" title="E69">A26</font></i>: 
( <font color="Maroon" title="c17">i'</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> &amp; <font color="Maroon" title="c17">i'</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>assume </b><a NAME="E15:96_3_1"/>
<font color="Maroon" title="c10">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">j</font>
 ;<br/>

<a NAME="E16:96_3_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E17:96_3_1"/><b>then </b>
<font color="Maroon" title="c17">i'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">j'</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E18:96_3_1"/><b>then </b>
( <font color="Maroon" title="c17">i'</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">j'</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  &amp; <font color="Maroon" title="c17">i'</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">j'</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T12" title="YELLOW_3:th.12">YELLOW_3:12</a></i>;<br/>
<a NAME="E19:96_3_1"/><b>then </b><i><font color="Green" title="E70">A27</font></i>: 
( <font color="Maroon" title="c14">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> &amp; <font color="Maroon" title="c15">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E8:96_3_1"><i><font color="Green" title="E66">A23</font></i></a>, <a class="txt" href="yellow_6.html#E10:96_3_1"><i><font color="Green" title="E67">A24</font></i></a>, <a class="txt" href="yellow_6.html#E14:96_3_1"><i><font color="Green" title="E69">A26</font></i></a>, <a class="ref" href="yellow_6.html#T40" target="_self" title="YELLOW_6:th.40">Th40</a></i>;<br/>
<a NAME="E20:96_3_1"/><i><font color="Green" title="E71">A28</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">W2</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E8:96"><i><font color="Green" title="E62">A19</font></i></a>, <a class="txt" href="yellow_6.html#E8:96_3_1"><i><font color="Green" title="E66">A23</font></i></a>, <a class="txt" href="yellow_6.html#E10:96_3_1"><i><font color="Green" title="E67">A24</font></i></a>, <a class="txt" href="yellow_6.html#E12:96_3_1"><i><font color="Green" title="E68">A25</font></i></a></i>;<br/>
<a NAME="E21:96_3_1"/>
<font color="Maroon" title="c14">W1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E19:96_3_1"><i><font color="Green" title="E70">A27</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E22:96_3_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E20:96_3_1"><i><font color="Green" title="E71">A28</font></i></a></i>;<br/>
<a NAME="E23:96_3_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E24:96_3_1"/>
<font color="Maroon" title="c8">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:96_3"/>
<font color="Maroon" title="c8">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c9">V</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:96_3"><i><font color="Green" title="E63">A20</font></i></a>, <a class="ref" href="waybel_0.html#T8" title="WAYBEL_0:th.8">WAYBEL_0:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:96"/><b>then </b><i><font color="Green" title="E63">A29</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c8">N</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D18" target="_self" title="YELLOW_6:def.18">Def18</a></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" title="c9">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">q</font>;<br/><a NAME="E1:96_4"/><i><font color="Green" title="E64">A30</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">V</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/><a NAME="E2:96_4"/>
<font color="Maroon" title="c8">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">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" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a>, <a class="ref" href="tops_1.html#T51" title="TOPS_1:th.51">TOPS_1:51</a></i>;<br/>
<a NAME="E2:96_4_1"/><b>then </b><i><font color="Green" title="E65">A31</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T39" target="_self" title="YELLOW_6:th.39">Th39</a></i>;<br/>
<a NAME="E3:96_4_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T53" title="TOPS_1:th.53">TOPS_1:53</a></i>;<br/>
<a NAME="E4:96_4_1"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T39" target="_self" title="YELLOW_6:th.39">Th39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">i</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">N</font> <b>by </b><i><a class="txt" href="yellow_6.html#E5:96"><i><font color="Green" title="E60">A14</font></i></a>, <a class="txt" href="yellow_6.html#E2:96_4_1"><i><font color="Green" title="E65">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">i</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon" title="c11">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">N</font>;<br/>

<b>consider </b><font color="Maroon" title="c12">j1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>, <font color="Maroon" title="c13">j2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span><b> such that </b><br/><a NAME="E7:96_4_1"/><i><font color="Green" title="E66">A32</font></i>: 
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c12">j1</font>,<font color="Maroon" title="c13">j2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E5:96"><i><font color="Green" title="E60">A14</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E8:96_4_1"/><i><font color="Green" title="E67">A33</font></i>: 
( <font color="Maroon" title="c11">j</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j1</font> &amp; <font color="Maroon" title="c11">j</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:96_4_1"><i><font color="Green" title="E66">A32</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">W1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E10:96_4_1"/><i><font color="Green" title="E68">A34</font></i>: 
( <font color="Maroon" title="c12">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">W1</font> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W1</font> &amp; <font color="Maroon" title="c14">W1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T38" target="_self" title="YELLOW_6:th.38">Th38</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">W2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E12:96_4_1"/><i><font color="Green" title="E69">A35</font></i>: 
( <font color="Maroon" title="c13">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">W2</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">W2</font> &amp; <font color="Maroon" title="c15">W2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T38" target="_self" title="YELLOW_6:th.38">Th38</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">j'</font> = <font color="Maroon" title="c11">j</font>, <font color="Maroon" title="c17">i'</font> = <font color="Maroon" title="c10">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> ;<br/>
<a NAME="E14:96_4_1"/><i><font color="Green" title="E70">A36</font></i>: 
( <font color="Maroon" title="c17">i'</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c17">i'</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>assume </b><a NAME="E15:96_4_1"/>
<font color="Maroon" title="c10">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">j</font>
 ;<br/>

<a NAME="E16:96_4_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c2">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">q</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E17:96_4_1"/><b>then </b>
<font color="Maroon" title="c17">i'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">j'</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E18:96_4_1"/><b>then </b>
( <font color="Maroon" title="c17">i'</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">j'</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  &amp; <font color="Maroon" title="c17">i'</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">j'</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T12" title="YELLOW_3:th.12">YELLOW_3:12</a></i>;<br/>
<a NAME="E19:96_4_1"/><b>then </b><i><font color="Green" title="E71">A37</font></i>: 
( <font color="Maroon" title="c14">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c15">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font> )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E8:96_4_1"><i><font color="Green" title="E67">A33</font></i></a>, <a class="txt" href="yellow_6.html#E12:96_4_1"><i><font color="Green" title="E69">A35</font></i></a>, <a class="txt" href="yellow_6.html#E14:96_4_1"><i><font color="Green" title="E70">A36</font></i></a>, <a class="ref" href="yellow_6.html#T40" target="_self" title="YELLOW_6:th.40">Th40</a></i>;<br/>
<a NAME="E20:96_4_1"/><i><font color="Green" title="E72">A38</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">W1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">W2</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E8:96"><i><font color="Green" title="E62">A19</font></i></a>, <a class="txt" href="yellow_6.html#E8:96_4_1"><i><font color="Green" title="E67">A33</font></i></a>, <a class="txt" href="yellow_6.html#E10:96_4_1"><i><font color="Green" title="E68">A34</font></i></a>, <a class="txt" href="yellow_6.html#E12:96_4_1"><i><font color="Green" title="E69">A35</font></i></a></i>;<br/>
<a NAME="E21:96_4_1"/>
<font color="Maroon" title="c14">W1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E19:96_4_1"><i><font color="Green" title="E71">A37</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E22:96_4_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E20:96_4_1"><i><font color="Green" title="E72">A38</font></i></a></i>;<br/>
<a NAME="E23:96_4_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E24:96_4_1"/>
<font color="Maroon" title="c8">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c9">V</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:96_4"/>
<font color="Maroon" title="c8">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c9">V</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:96_4"><i><font color="Green" title="E64">A30</font></i></a>, <a class="ref" href="waybel_0.html#T8" title="WAYBEL_0:th.8">WAYBEL_0:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:96"/><b>then </b>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c8">N</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D18" target="_self" title="YELLOW_6:def.18">Def18</a></i>;<br/>
<b>hence </b><a NAME="E14:96"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow_6.html#E2:96"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="yellow_6.html#E3:96"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="yellow_6.html#E11:96"><i><font color="Green" title="E63">A29</font></i></a></i>;<br/>


</div>
