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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#L1">TopStruct</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">P</font> be   <a href="funct_2.html#NM2">Permutation</a> of  <a href="numbers.html#K5">NAT</a> ;<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>assume </b><a NAME="E1:58"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">S</font> <a href="frechet.html#R1">is_convergent_to</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:58"/>
for <font color="Olive">U1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">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="Olive">U1</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="frechet2.html#K1" target="_self">*</a> <font color="Maroon">P</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Olive">U1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">U1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:58_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:58_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>;<br/>
<a NAME="E3:58_1"/><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

<b>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="E2:58_1_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> <font color="Maroon">U1</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E8"><i><font color="Green">Lemma45</font></i></a>, , <a class="txt" href="frechet2.html#E9"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="frechet.html#D2">FRECHET:def 2</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>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E3:58_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E4:58_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font>
 ;<br/>

<b>thus </b><a NAME="E5:58_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 <b>by </b><i><a class="ref" href="frechet2.html#T3" target="_self">Th3</a>, <a class="ref" href="frechet2.html#T4" target="_self">Th4</a>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:58_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>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="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="frechet2.html#K1" target="_self">*</a> <font color="Maroon">P</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font>]
 <b>from </b><i><a class="ref" href="frechet2.html#S4" target="_self">FRECHET2:sch 4</a>(<a class="txt" href="frechet2.html#E11"><i><font color="Green">Lemma48</font></i></a>);</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:58"/>
<font color="Maroon">S</font> <a href="frechet2.html#K1" target="_self">*</a> <font color="Maroon">P</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#D2">FRECHET:def 2</a></i>;<br/>


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