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<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>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:80"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">P</font> is <a href="jordan2c.html#V1">Bounded</a>
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 ;<br/>

<b>assume </b><a NAME="E3:80"/><i><font color="Green">E54</font></i>: 
not  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">S</font> is <a href="jordan2c.html#V1">Bounded</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> as  <a href="tbsp_1.html#V6">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="txt" href="kurato_2.html#E2:80"><i><font color="Green">E49</font></i></a>, <a class="ref" href="jordan2c.html#D2">JORDAN2C:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">z</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E6:80"/><i><font color="Green">E56</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font> &amp; <font color="Maroon">P'</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">t</font> )
 <b>by </b><i><a class="ref" href="metric_6.html#D10">METRIC_6:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">r</font> = 1;<br/>
<b>set </b><font color="Maroon">R</font> = <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span>;<br/>
<a NAME="E7:80"/>
3 <a href="real_1.html#K4">*</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E8:80"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E10:80"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">S</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">S</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">k1</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="E12:80"/><i><font color="Green">E72</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="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k1</font> holds <br/><font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,1
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">k2</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="E14:80"/><i><font color="Green">E77</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="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font> holds <br/><font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,1
 <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E15:80"/>
<font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P'</font>
 
<b>by </b><i><a class="ref" href="kurato_2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E16:80"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E17:80"/>
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k1</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E18:80"/><b>then </b>
<span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E19:80"/><b>then </b>
<font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,1
 
<b>by </b><i><a class="txt" href="kurato_2.html#E3:80"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E20:80"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">t</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,1
 
<b>by </b><i><a class="txt" href="kurato_2.html#E6:80"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E21:80"/>
 <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<a NAME="E22:80"/><b>then </b>
<span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E23:80"/><b>then </b>
<font color="Maroon">S</font> <a href="kurato_2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,1
 
<b>by </b><i/>;<br/>
<a NAME="E24:80"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">t</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,1
 
<b>by </b><i><a class="txt" href="kurato_2.html#E6:80"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E25:80"/>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> 3 <a href="real_1.html#K4">*</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E26:80"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E27:80"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E10:80"><i><font color="Green">E57</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E28:80"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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