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<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>let </b><font color="Maroon">F</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>, <font color="Maroon">G</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:75"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">G</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:75"/>
 <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">F</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:75_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">G</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:75_1"/>
for <font color="Olive">H</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>;<br/>

<b>consider </b><font color="Maroon">H1</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:75_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">H1</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font> &amp; <font color="Maroon">H1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#T7">CONNSP_2:7</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:75_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">G</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">H1</font>
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E5:75_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E6:75_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">H1</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E1:75_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:75_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="kurato_2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E10:75_1_1"/>
<font color="Maroon">H1</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="kurato_2.html#T4" target="_self">Th4</a>, <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">H'</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E12:75_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">H'</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">y</font> &amp; <font color="Maroon">H'</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">H'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#T7">CONNSP_2:7</a></i>;<br/>
<a NAME="E13:75_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="kurato_2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E14:75_1_1"/><b>then </b>
<font color="Maroon">H'</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="connsp_2.html#T34">CONNSP_2:34</a></i>;<br/>
<a NAME="E15:75_1_1"/><b>then </b>
<font color="Maroon">H1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<b>hence </b><a NAME="E16:75_1_1"/>
<font color="Maroon">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:75_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E3:75"/>
 <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2"><b>proof </b></a><div class="add">

<a NAME="E1:75_2"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:75_2_1"/>
<font color="Maroon">G</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:75_2_1"/>
<font color="Maroon">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:75_2"/>
 <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#T3" target="_self">Th3</a></i>;<br/>


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


</div>
