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<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<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="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<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="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">S</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">St</font> &amp; <font color="Maroon">St</font> is <a href="frechet.html#V2">convergent</a> )
 ;<br/>

<a NAME="E2:41"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S</font> is <a href="frechet.html#V2">convergent</a>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Sn</font> = <font color="Maroon">S</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font> ;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" target="_self">TopSpaceNorm</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E5:41"/><i><font color="Green">E42</font></i>: 
<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="ref" href="frechet.html#D3">FRECHET:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">xxt</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E7:41"/>
<font color="Maroon">St</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">xxt</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then </b><i><font color="Green">E43</font></i>:  <a href="frechet2.html#K4">lim</a> <font color="Maroon">St</font> = 
<font color="Maroon">xxt</font>
<b>by </b><i><a class="ref" href="frechet2.html#T28">FRECHET2:28</a></i>
<br/>.= 
 <a href="frechet2.html#K4">lim</a> <font color="Maroon">S</font>
<b>by </b><i>, <a class="ref" href="frechet2.html#T28">FRECHET2:28</a></i>
;<br/>
<b>consider </b><font color="Maroon">xt</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" target="_self">LinearTopSpaceNorm</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E10:41"/><i><font color="Green">E44</font></i>: 
 <a href="frechet.html#K1">Lim</a> <font color="Maroon">St</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">xt</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="frechet2.html#T27">FRECHET2:27</a>, </i>;<br/>
<a NAME="E11:41"/>
<font color="Maroon">xt</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">xt</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:41"/><b>then </b>
<font color="Maroon">St</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">xt</font>
 
<b>by </b><i>, <a class="ref" href="frechet.html#D4">FRECHET:def 4</a></i>;<br/>
<b>then </b> <a href="frechet.html#K1">Lim</a> <font color="Maroon">St</font> = 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="frechet2.html#K4">lim</a> <font color="Maroon">St</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, <a class="ref" href="frechet2.html#T28">FRECHET2:28</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8">lim</a> <font color="Maroon">Sn</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E14:41"/>
(  <a href="frechet.html#K1">Lim</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="frechet.html#K1">Lim</a> <font color="Maroon">St</font> &amp;  <a href="frechet2.html#K4">lim</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="frechet2.html#K4">lim</a> <font color="Maroon">St</font> )
 <b>by </b><i/>;<br/>


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