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<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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1"/>
( <font color="Maroon">S</font> is <a href="frechet.html#V2">convergent</a> or  not <font color="Maroon">S</font> is <a href="frechet.html#V2">convergent</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>suppose </b></a><a NAME="E1:25_1_1"/>
<font color="Maroon">S</font> is <a href="frechet.html#V2">convergent</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1"/>
 <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="frechet.html#T26">FRECHET:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2"><b>suppose </b></a><a NAME="E1:25_1_2"/>
not <font color="Maroon">S</font> is <a href="frechet.html#V2">convergent</a>
 ;<br/><div class="add"><a NAME="E2:25_1_2"/><b>then </b>
 <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:25_1_2"/>
 <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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