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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">F</font> being <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <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>)</span> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c1">T</font> iff  <a href="yellow19.html#K1" title="YELLOW19:func.1">NeighborhoodSystem</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">F</font> )</span><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <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>)</span> holds <br/> ( <font color="Maroon" title="c2">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c1">T</font> iff  <a href="yellow19.html#K1" title="YELLOW19:func.1">NeighborhoodSystem</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> )</span><br/>



<a NAME="E1:5"/><i><font color="Green" title="E4">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <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="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">F</font> be  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <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>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">T</font> iff  <a href="yellow19.html#K1" title="YELLOW19:func.1">NeighborhoodSystem</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow19.html#K1" title="YELLOW19:func.1">NeighborhoodSystem</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> implies <font color="Maroon" title="c2">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">T</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:5_1"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow19.html#K1" title="YELLOW19:func.1">NeighborhoodSystem</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font></span><br/><b>thus </b><a NAME="E2:5_1"/>
 <a href="yellow19.html#K1" title="YELLOW19:func.1">NeighborhoodSystem</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow19.html#K1" title="YELLOW19:func.1">NeighborhoodSystem</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<b>assume </b><a NAME="E1:5_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow19.html#K1" title="YELLOW19:func.1">NeighborhoodSystem</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> <b>by </b><i><a class="ref" href="yellow19.html#T3" target="_self" title="YELLOW19:th.3">Th3</a></i>;<br/>
<a NAME="E3:5_1_1"/>
<font color="Maroon" title="c2">x</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="c5">y</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E4:5_1_1"/><b>then </b>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y</font> )
 
<b>by </b><i><a class="txt" href="yellow19.html#E1:5_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a>, <a class="ref" href="waybel_7.html#D5" title="WAYBEL_7:def.5">WAYBEL_7:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:5_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:5"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="waybel_7.html#T11" title="WAYBEL_7:th.11">WAYBEL_7:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E3:5"/><i><font color="Green" title="E5">A3</font></i>: 
 <a href="yellow19.html#K1" title="YELLOW19:func.1">NeighborhoodSystem</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">T</font></span><br/>

<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D5" title="WAYBEL_7:def.5">WAYBEL_7:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> or <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<b>assume </b><a NAME="E4:5"/>
( <font color="Maroon" title="c4">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font></span><br/>

<a NAME="E5:5"/><b>then </b>
<font color="Maroon" title="c4">A</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T5" title="CONNSP_2:th.5">CONNSP_2:5</a></i>;<br/>
<a NAME="E6:5"/><b>then </b>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow19.html#K1" title="YELLOW19:func.1">NeighborhoodSystem</a> <font color="Maroon" title="c2">x</font>
 
;<br/>
<b>hence </b><a NAME="E7:5"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E3:5"><i><font color="Green" title="E5">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
