<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/>





<b>given </b><font color="Maroon" title="c4">S1</font> being    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c2">S</font><b> such that </b><a NAME="E1:49"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c4">S1</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c3">x</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">O</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="frechet2.html#D4" target="_self" title="FRECHET2:def.4">FRECHET2:def 4</a><br/><b>let </b><font color="Maroon" title="c6">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:49"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c5">O</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E3:49"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:49"/><i><font color="Green" title="E42">A4</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c4">S1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1:49"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="frechet2.html#E2:49"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="frechet2.html#E3:49"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="frechet.html#D2" title="FRECHET:def.2">FRECHET:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">n2</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c6">n</font>;<br/>
<b>consider </b><font color="Maroon" title="c9">NS</font> being  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E7:49"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c4">S1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">S</font> <a href="frechet2.html#K2" title="FRECHET2:func.2">*</a> <font color="Maroon" title="c9">NS</font>
 <b>by </b><i><a class="ref" href="bhsp_3.html#D4" title="BHSP_3:def.4">BHSP_3:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">NS</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c6">n</font></span>)</span>
;<br/>

<a NAME="E8:49"/><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c7">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E9:49"/><i><font color="Green" title="E45">A7</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E10:49"/>
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">NS</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c6">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seqm_3.html#T33" title="SEQM_3:th.33">SEQM_3:33</a></i>;<br/>
<b>hence </b><a NAME="E11:49"/>
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">NS</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c6">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="frechet2.html#E9:49"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>

<a NAME="E12:49"/><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c4">S1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E5:49"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="frechet2.html#E8:49"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/>
<a NAME="E13:49"/>
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E14:49"/><b>then </b>
 <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">NS</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#T3" title="SEQ_1:th.3">SEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E15:49"/>
<font color="Maroon" title="c2">S</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">NS</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c7">n1</font>,<font color="Maroon" title="c6">n</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E7:49"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="frechet2.html#E12:49"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>


</div>
