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<a NAME="E1:97"/>
<span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default">1</span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span> <a href="subset_1.html#R1">misses</a> <span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default">1</span><a href="borsuk_5.html#NK2" target="_self">.[</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E2:97"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">4</span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">5</span><a href="seq_4.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default">1</span><a href="borsuk_5.html#NK2" target="_self">.[</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E114"><i><font color="Green">Lemma78</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E3:97"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default">1</span><a href="borsuk_5.html#NK2" target="_self">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:97_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default">1</span><a href="borsuk_5.html#NK2" target="_self">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:97_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:97_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="borsuk_5.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E4:97_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:97_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T16" target="_self">Th16</a></i>;<br/>


</div><b>end;</b></div>
<span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default">1</span><a href="borsuk_5.html#NK2" target="_self">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="seq_4.html#K1">{</a><span class="default">4</span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">5</span><a href="seq_4.html#K1">}</a></span></span>)</span> = 
<span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default">1</span><a href="borsuk_5.html#NK2" target="_self">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p4"><a href="seq_4.html#K1">{</a><span class="default">4</span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="seq_4.html#K1">{</a><span class="default">5</span><a href="seq_4.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T113">XBOOLE_1:113</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default">1</span><a href="borsuk_5.html#NK2" target="_self">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span>
<b>by </b><i><a class="txt" href="borsuk_5.html#E2:97"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_1.html#T78">XBOOLE_1:78</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default">1</span><a href="borsuk_5.html#NK2" target="_self">.[</a></span>
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:97"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E5:97"/>
<span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default">1</span><a href="borsuk_5.html#NK2" target="_self">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default">2</span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> 2,4</span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="seq_4.html#K1">{</a><span class="default">4</span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default">5</span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default">1</span><a href="borsuk_5.html#NK2" target="_self">.[</a></span>
 ;<br/>


</div>
