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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> holds <br/> <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Olive">r</font> <a href="metric_6.html#R2" target="_self">contains_almost_all_sequence</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:23"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:23_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:23_1"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="metric_6.html#R2" target="_self">contains_almost_all_sequence</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:23_1"/>
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:23"/>
for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">X</font> holds <br/><font color="Olive">V</font> <a href="metric_6.html#R2" target="_self">contains_almost_all_sequence</a> <font color="Maroon">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:23_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E2:23_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">X</font>
 ;<br/>

<b>consider </b><font color="Maroon">q</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:23_2"/><i><font color="Green">E44</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i>, , <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">m1</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="E6:23_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:23_2_1"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:23_2_1"/><b>then </b>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">q</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:23_2_1"/>
<font color="Maroon">S</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:23_2"/>
<font color="Maroon">V</font> <a href="metric_6.html#R2" target="_self">contains_almost_all_sequence</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


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