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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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="struct_0.html#NM9" title="STRUCT_0:NM.9">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>let </b><font color="Maroon" title="c4">S1</font> be    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c2">S</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="frechet2.html#R1" title="FRECHET2:pred.1">is_a_cluster_point_of</a> <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green" title="E48">A2</font></i>: 
 ex <font color="Olive" title="b1">n0</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 <br/>for <font color="Olive" title="b2">n</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>  holds  <font color="Maroon" title="c4">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n0</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">n0</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="E4:60"/><i><font color="Green" title="E49">A3</font></i>: 
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon" title="c4">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n0</font></span>)</span>
 <b>by </b><i><a class="txt" href="frechet2.html#E2:60"><i><font color="Green" title="E48">A2</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">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="c7">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="E5:60"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c6">O</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E6:60"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">m0</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="E8:60"/><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m0</font>
 <b>and </b><br/><a NAME="E9:60"/><i><font color="Green" title="E53">A7</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E1:60"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="frechet2.html#E5:60"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="frechet2.html#E6:60"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="frechet2.html#D4" target="_self" title="FRECHET2:def.4">Def4</a></i>;<br/>
<a NAME="E10:60"/>
<font color="Maroon" title="c5">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">n0</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E11:60"/><b>then </b>
<font color="Maroon" title="c5">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m0</font>
 
<b>by </b><i><a class="txt" href="frechet2.html#E8:60"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c8">m0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">n0</font> as    <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>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">m</font>
;<br/>

<b>thus </b><a NAME="E13:60"/>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E8:60"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>

<font color="Maroon" title="c4">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font> = 
<font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">m0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">n0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">n0</font></span>)</span>
<b>by </b><i><a class="txt" href="frechet2.html#E4:60"><i><font color="Green" title="E49">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m0</font>

;<br/>
<b>hence </b><a NAME="E15:60"/>
<font color="Maroon" title="c4">S1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 <b>by </b><i><a class="txt" href="frechet2.html#E9:60"><i><font color="Green" title="E53">A7</font></i></a></i>;<br/>


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