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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>





<b>assume </b><a NAME="E1:60"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="borsuk_5.html#T62">BORSUK_5:62</a></i>;<br/>
<a NAME="E3:60"/><i><font color="Green">E45</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>thus </b><a NAME="E4:60"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:60_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3">R^1</a>  ;<br/>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E2:60_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tops_1.html#K2">Fr</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="tops_1.html#T73">TOPS_1:73</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:60_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <a href="topalg_2.html#K3">R^1</a> 
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:60"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rcomp_1.html#T13">RCOMP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:60_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:60_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tops_1.html#T47">TOPS_1:47</a></i>;<br/></div><b>end;</b></div><a NAME="E4:60_1_1"/><b>then </b>
 <a href="tops_1.html#K2">Fr</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="rcomp_3.html#E1:60"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rcomp_3.html#T44" target="_self">Th44</a></i>;<br/><a NAME="E5:60_1_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2">Fr</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K2">Fr</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E6:60_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rcomp_3.html#E2:60_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rcomp_3.html#E1:60_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rcomp_3.html#E1:60_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:60_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:60"><i><font color="Green">E44</font></i></a>, <a class="txt" href="rcomp_3.html#E1:60_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="rcomp_3.html#E3:60"><i><font color="Green">E45</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E5:60_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E3:60_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:60_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


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

<a NAME="E5:60"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topreala.html#T19">TOPREALA:19</a></i>;<br/>
<a NAME="E6:60"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rcomp_3.html#E1:60"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>
<b>hence </b><a NAME="E7:60"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div>
