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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:106"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK2" target="_self">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_2.html#K2">.]</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="borsuk_5.html#K3" target="_self">IRRAT</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="seq_4.html#K1">}</a></span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK2" target="_self">.[</a></span>, <font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_2.html#K2">.]</a></span>, <font color="Maroon">c<sub>7</sub></font> =  <a href="borsuk_5.html#K3" target="_self">IRRAT</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="seq_4.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E3:106"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>4</sub></font></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>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:106"><i><font color="Green">E86</font></i></a>, <a class="ref" href="pre_topc.html#T50">PRE_TOPC:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="borsuk_5.html#T61" target="_self">Th61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="pre_topc.html#T50">PRE_TOPC:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:106"><i><font color="Green">E86</font></i></a>, <a class="ref" href="borsuk_5.html#T97" target="_self">Th97</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:106"><i><font color="Green">E86</font></i></a>, <a class="ref" href="borsuk_5.html#T55" target="_self">Th55</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_5.html#NK1" target="_self">.]</a></span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:106"><i><font color="Green">E86</font></i></a>, <a class="ref" href="borsuk_5.html#T29" target="_self">Th29</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_5.html#NK1" target="_self">.]</a></span>
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:106"><i><font color="Green">E87</font></i></a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>
<b>hence </b><a NAME="E5:106"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_5.html#NK1" target="_self">].-infty,</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_5.html#NK1" target="_self">.]</a></span>
 ;<br/>


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