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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">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">S'</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>let </b><font color="Maroon">x'</font> be   <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>;<br/>









<b>assume </b><a NAME="E1:45"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">S</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">S'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> )
 ;<br/>

<b>thus </b><a NAME="E2:45"/>
( <font color="Maroon">S</font> <a href="metric_6.html#R1">is_convergent_in_metrspace_to</a> <font color="Maroon">x</font> implies <font color="Maroon">S'</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x'</font> )
 <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"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S</font> <a href="metric_6.html#R1">is_convergent_in_metrspace_to</a> <font color="Maroon">x</font>
 ;<br/>

<b>let </b><font color="Maroon">U1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet.html#D2">FRECHET:def 2</a><br/>

<b>assume </b><a NAME="E2:45_1"/>
( <font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:45_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:45_1"/><i><font color="Green">E53</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E5"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E7:45_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="metric_6.html#R2">contains_almost_all_sequence</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="frechet2.html#T1" target="_self">Th1</a>, <a class="ref" href="metric_6.html#T30">METRIC_6:30</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:45_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_6.html#D12">METRIC_6:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E10:45_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E11:45_1"/><b>then </b>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:45_1"/>
<font color="Maroon">S'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E5"><i><font color="Green">Lemma37</font></i></a>, </i>;<br/>


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

<b>assume </b><a NAME="E3:45"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">S'</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x'</font>
 ;<br/>

<a NAME="E4:45"/>
for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">M</font>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">M</font> holds <br/><font color="Olive">V</font> <a href="metric_6.html#R2">contains_almost_all_sequence</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E2:45_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">M</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">V'</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</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/>
<b>reconsider </b><font color="Maroon">V'</font> = <font color="Maroon">V'</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> ;<br/>
<a NAME="E5:45_2"/>
<font color="Maroon">V'</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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="E6:45_2"/><b>then </b>
<font color="Maroon">V'</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:45_2"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S'</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V'</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E5"><i><font color="Green">Lemma37</font></i></a>, , <a class="ref" href="frechet2.html#T1" target="_self">Th1</a>, <a class="ref" href="frechet.html#D2">FRECHET:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_6.html#D12">METRIC_6:def 12</a><br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E9:45_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>hence </b><a NAME="E10:45_2"/>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E5"><i><font color="Green">Lemma37</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:45"/>
<font color="Maroon">S</font> <a href="metric_6.html#R1">is_convergent_in_metrspace_to</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="metric_6.html#T32">METRIC_6:32</a></i>;<br/>


</div>
