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<span class="kw">set </span><font color="Maroon" title="c3">B</font> =  <a href="frechet.html#K2" title="FRECHET:func.2">Balls</a> <font color="Maroon" title="c2">x</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">x9</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E2:15_1_1"/><span class="lab"><font color="Green" title="E12">A0</font></span>: 
( <font color="Maroon" title="c4">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="frechet.html#K2" title="FRECHET:func.2">Balls</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">x9</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="frechet.html#D1" target="_self" title="FRECHET:def.1">Def10</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <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> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K10</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>) =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">x9</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">M</font></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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> ] <span class="kw">means </span><font color="Maroon" title="c1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E3:15_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] </span>}</span>  is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="card_4.html#S1" title="CARD_4:sch.1">CARD_4:sch 1</a>();<br/></span>
<span class="kw">hence </span><a NAME="E4:15_1_1"/>
 <a href="frechet.html#K2" title="FRECHET:func.2">Balls</a> <font color="Maroon" title="c2">x</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1"><span class="lab"><font color="Green" title="E12">A0</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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