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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><a NAME="E1:45"/><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>
 ;<br/>

<b>thus </b><a NAME="E2:45"/>
( <font color="Maroon" title="c2">Sm</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a> implies <font color="Maroon" title="c3">St</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon" title="c2">Sm</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">Sm</font> <a href="metric_6.html#R1" title="METRIC_6:pred.1">is_convergent_in_metrspace_to</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="metric_6.html#T22" title="METRIC_6:th.22">METRIC_6:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</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="E5:45_1"/>
<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#E1:45"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="frechet2.html#E3:45_1"><i><font color="Green" title="E37">A2</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="E6:45_1"/>
<font color="Maroon" title="c3">St</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>
 <b>by </b><i><a class="ref" href="frechet.html#D3" title="FRECHET:def.3">FRECHET:def 3</a></i>;<br/>


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

<b>assume </b><a NAME="E3:45"/>
<font color="Maroon" title="c3">St</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x'</font> being   <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> such that </b><br/><a NAME="E5:45"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c3">St</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c4">x'</font>
 <b>by </b><i><a class="ref" href="frechet.html#D3" title="FRECHET:def.3">FRECHET:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x'</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E7:45"/>
<font color="Maroon" title="c2">Sm</font> <a href="metric_6.html#R1" title="METRIC_6:pred.1">is_convergent_in_metrspace_to</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E1:45"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="frechet2.html#E5:45"><i><font color="Green" title="E37">A3</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="E8:45"/>
<font color="Maroon" title="c2">Sm</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 <b>by </b><i><a class="ref" href="metric_6.html#T21" title="METRIC_6:th.21">METRIC_6:21</a></i>;<br/>


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