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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"> ( <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> iff 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>  st <font color="Olive" title="b1">N</font> is <a href="yellow19.html#V1" title="YELLOW19:attr.1">Cauchy</a> holds <br/><font color="Olive" title="b1">N</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a> )</span><br/>

<b>thus </b><a NAME="E1:49"/>
( <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</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>  st <font color="Olive" title="b1">N</font> is <a href="yellow19.html#V1" title="YELLOW19:attr.1">Cauchy</a> holds <br/><font color="Olive" title="b1">N</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</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">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">N</font> is <a href="yellow19.html#V1" title="YELLOW19:attr.1">Cauchy</a> holds <br/><font color="Olive" title="b1">N</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a> ) implies <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</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">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">N</font> is <a href="yellow19.html#V1" title="YELLOW19:attr.1">Cauchy</a> holds <br/><font color="Olive" title="b1">N</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">N</font> is <a href="yellow19.html#V1" title="YELLOW19:attr.1">Cauchy</a> implies <font color="Maroon" title="c2">N</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a> )</span><br/>

<b>assume </b><a NAME="E2:49_1"/><i><font color="Green" title="E29">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  holds <br/>( <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b1">A</font> or <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow19.html#D5" target="_self" title="YELLOW19:def.5">YELLOW19:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">N</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a></span><br/>

<b>consider </b><font color="Maroon" title="c3">x</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><br/><a NAME="E4:49_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font>
 <b>by </b><i><a class="txt" href="#E1:49_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="yellow19.html#E37"><i><font color="Green" title="E24">Lm1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">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="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c4">V</font></span><br/><b>assume </b><a NAME="E1:49_1_1"/>
not <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="c4">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:49_1_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="c4">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E2:49_1"><i><font color="Green" title="E29">A2</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E4:49_1_1"/><i><font color="Green" title="E31">A4</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a></i>;<br/><a NAME="E5:49_1_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c4">V</font>
 <b>by </b><i><a class="txt" href="#E4:49_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="waybel_9.html#D9" title="WAYBEL_9:def.9">WAYBEL_9:def 9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E7:49_1_1"/><i><font color="Green" title="E32">A5</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</a></i>;<br/><a NAME="E8:49_1_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E4:49_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="#E7:49_1_1"><i><font color="Green" title="E32">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E9:49_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:49_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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>
<a NAME="E6:49_1"/><b>then </b>
<font color="Maroon" title="c3">x</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>by </b><i><a class="ref" href="yellow_6.html#D18" title="YELLOW_6:def.18">YELLOW_6:def 18</a></i>;<br/>
<b>hence </b><a NAME="E7:49_1"/>
<font color="Maroon" title="c2">N</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a>
 <b>by </b><i><a class="ref" href="yellow_6.html#D19" title="YELLOW_6:def.19">YELLOW_6:def 19</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>

<b>assume </b><a NAME="E2:49"/><i><font color="Green" title="E28">A6</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>  st <font color="Olive" title="b1">N</font> is <a href="yellow19.html#V1" title="YELLOW19:attr.1">Cauchy</a> holds <br/><font color="Olive" title="b1">N</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">F</font> be  <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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">  ex <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> st <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="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">T</font></span><br/><b>consider </b><font color="Maroon" title="c3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c2">F</font></span>)</span>;<br/><a NAME="E2:49_2"/>
 <a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c2">F</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a>
 <b>by </b><i><a class="txt" href="#E2:49"><i><font color="Green" title="E28">A6</font></i></a></i>;<br/><a NAME="E3:49_2"/><b>then </b><i><font color="Green" title="E29">A7</font></i>: 
 <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c2">F</font></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="yellow_6.html#D19" title="YELLOW_6:def.19">YELLOW_6:def 19</a></i>;<br/><a NAME="E4:49_2"/><b>then </b>
<font color="Maroon" title="c3">x</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> <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c2">F</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/><b>take </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">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="c5">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">T</font></span><br/><b>thus </b><a NAME="E6:49_2"/>
<font color="Maroon" title="c5">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="#E3:49_2"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="yellow19.html#T18" target="_self" title="YELLOW19:th.18">Th18</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>
<b>hence </b><a NAME="E4:49"/>
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>
 <b>by </b><i><a class="ref" href="yellow19.html#T33" target="_self" title="YELLOW19:th.33">Th33</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>
