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<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">Sm</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">St</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:47"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">Sm</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">St</font>
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">Sm</font> is <a href="tbsp_1.html#V2">convergent</a>
 ;<br/>

<a NAME="E3:47"/><i><font color="Green">E32</font></i>: 
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T38">PCOMPS_1:38</a></i>;<br/>
<b>set </b><font color="Maroon">x</font> =  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">Sm</font>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> =  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">Sm</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E5:47"/>
<font color="Maroon">Sm</font> <a href="metric_6.html#R1">is_convergent_in_metrspace_to</a>  <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">Sm</font>
 
<b>by </b><i>, <a class="ref" href="metric_6.html#T27">METRIC_6:27</a></i>;<br/>
<a NAME="E6:47"/><b>then </b>
<font color="Maroon">St</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E5"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="frechet2.html#E4"><i><font color="Green">Lemma36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:47"/>
 <a href="tbsp_1.html#K1">lim</a> <font color="Maroon">Sm</font> <a href="hidden.html#R1">=</a>  <a href="frechet2.html#K4" target="_self">lim</a> <font color="Maroon">St</font>
 <b>by </b><i><a class="ref" href="frechet2.html#T1" target="_self">Th1</a>, </i>;<br/>


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