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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">Sm</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c3">St</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c2">Sm</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">St</font>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">Sm</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 ;<br/>

<a NAME="E3:46"/><i><font color="Green" title="E38">A3</font></i>: 
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a> 
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T38" title="PCOMPS_1:th.38">PCOMPS_1:38</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">x</font> =  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c2">Sm</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x'</font> =  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c2">Sm</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E5:46"/>
<font color="Maroon" title="c2">Sm</font> <a href="metric_6.html#R1" title="METRIC_6:pred.1">is_convergent_in_metrspace_to</a>  <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c2">Sm</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E2:46"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="metric_6.html#T27" title="METRIC_6:th.27">METRIC_6:27</a></i>;<br/>
<a NAME="E6:46"/><b>then </b>
<font color="Maroon" title="c3">St</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c5">x'</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E1:46"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="frechet2.html#T31" target="_self" title="FRECHET2:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E7:46"/>
 <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c2">Sm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="frechet2.html#K4" title="FRECHET2:func.4">lim</a> <font color="Maroon" title="c3">St</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E3:46"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="frechet2.html#T28" target="_self" title="FRECHET2:th.28">Th28</a></i>;<br/>


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