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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>assume </b><a NAME="E1:42"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">T</font> <a href="compts_1.html#NR2" target="_self">is_T2</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>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>



<b>thus </b><a NAME="E2:42"/>
( <font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x</font> implies ( <font color="Maroon">S</font> is <a href="frechet.html#V2">convergent</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="frechet2.html#K4" target="_self">lim</a> <font color="Maroon">S</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x</font>
 ;<br/>

<b>hence </b><a NAME="E2:42_1"/>
<font color="Maroon">S</font> is <a href="frechet.html#V2">convergent</a>
 <b>by </b><i><a class="ref" href="frechet.html#D3">FRECHET:def 3</a></i>;<br/>

<a NAME="E3:42_1"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st  <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">y</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="frechet2.html#E4"><i><font color="Green">Lemma36</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:42_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="frechet2.html#K4" target="_self">lim</a> <font color="Maroon">S</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#E3"><i><font color="Green">Lemma33</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E3:42"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">S</font> is <a href="frechet.html#V2">convergent</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="frechet2.html#K4" target="_self">lim</a> <font color="Maroon">S</font> )
 ;<br/>

<a NAME="E4:42"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> st  <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="frechet2.html#E4"><i><font color="Green">Lemma36</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:42"/>
<font color="Maroon">S</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#E3"><i><font color="Green">Lemma33</font></i></a></i>;<br/>


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