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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</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="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E16">A1</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">St</font> &amp; <font color="Maroon" title="c3">St</font> is <a href="frechet.html#V2" title="FRECHET:attr.2">convergent</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">Sm</font> = <font color="Maroon" title="c3">St</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K2" title="NORMSP_2:func.2">MetricSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<a NAME="E3:21"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c4">Sm</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 
<b>by </b><i><a class="txt" href="normsp_2.html#E1:21"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="frechet2.html#T32" title="FRECHET2:th.32">FRECHET2:32</a></i>;<br/>
<b>then </b><i><font color="Green" title="E18">A3</font></i>:  <a href="frechet2.html#K4" title="FRECHET2:func.4">lim</a> <font color="Maroon" title="c3">St</font> = 
 <a href="tbsp_1.html#K1" title="TBSP_1:func.1">lim</a> <font color="Maroon" title="c4">Sm</font>
<b>by </b><i><a class="ref" href="frechet2.html#T33" title="FRECHET2:th.33">FRECHET2:33</a></i>
<br/>.= 
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">S</font>
<b>by </b><i><a class="txt" href="normsp_2.html#E1:21"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="normsp_2.html#E3:21"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="normsp_2.html#T6" target="_self" title="NORMSP_2:th.6">Th6</a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c5">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="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span><b> such that </b><br/><a NAME="E6:21"/><i><font color="Green" title="E19">A4</font></i>: 
 <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c3">St</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="normsp_2.html#E1:21"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="frechet2.html#T27" title="FRECHET2:th.27">FRECHET2:27</a></i>;<br/>
<a NAME="E7:21"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c3">St</font>
 
<b>by </b><i><a class="txt" href="normsp_2.html#E6:21"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:21"/><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="ref" href="frechet.html#D4" title="FRECHET:def.4">FRECHET:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:21"/>
 <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c3">St</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="normsp_2.html#E4:21"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="normsp_2.html#E6:21"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="frechet2.html#T28" title="FRECHET2:th.28">FRECHET2:28</a></i>;<br/>

<b>thus </b><a NAME="E10:21"/>
 <a href="frechet2.html#K4" title="FRECHET2:func.4">lim</a> <font color="Maroon" title="c3">St</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="normsp_2.html#E4:21"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/>


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