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<div class="add">

<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>;<br/>

<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="frechet.html#V3" title="FRECHET:attr.3">Frechet</a>
 ;<br/>

<a NAME="E2:36"/>
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>  st ( for <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">S</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> holds <br/> <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) holds <br/><font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green" title="E33">A2</font></i>: 
for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">S</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> holds <br/> <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E2:36_1"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E3:36_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</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><br/>

<b>assume </b><a NAME="E1:36_1_1"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">p</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>consider </b><font color="Maroon" title="c5">S</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:36_1_1"/><i><font color="Green" title="E36">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E5:36_1_1"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c5">S</font>
 <b>by </b><i><a class="txt" href="frechet.html#E1:36"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="frechet.html#E1:36_1_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="frechet.html#D5" target="_self" title="FRECHET:def.5">Def5</a></i>;<br/>
<a NAME="E6:36_1_1"/>
<font color="Maroon" title="c5">S</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E5:36_1_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="frechet.html#D4" target="_self" title="FRECHET:def.4">Def4</a></i>;<br/>
<a NAME="E7:36_1_1"/><b>then </b>
<font color="Maroon" title="c5">S</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>
 
<b>by </b><i><a class="ref" href="frechet.html#D3" target="_self" title="FRECHET:def.3">Def3</a></i>;<br/>
<a NAME="E8:36_1_1"/><b>then </b>
 <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E1:36_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="frechet.html#E4:36_1_1"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:36_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="frechet.html#E5:36_1_1"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:36_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="frechet.html#E2:36_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:36"/>
<font color="Maroon" title="c1">T</font> is <a href="frechet.html#V4" title="FRECHET:attr.4">sequential</a>
 <b>by </b><i><a class="ref" href="frechet.html#T27" target="_self" title="FRECHET:th.27">Th27</a></i>;<br/>


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