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<b>let </b><font color="Maroon">A</font> be  <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</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="E1:116"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:116_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> or ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 </span><b>by </b><i><a class="ref" href="borsuk_5.html#T37" target="_self">Th37</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1"><b>suppose </b></a><a NAME="E1:116_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:116_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T37" target="_self">Th37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_2"><b>suppose </b></a><a NAME="E1:116_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><div class="add"><b>assume </b><a NAME="E2:116_1_2"/>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E3:116_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="zfmisc_1.html#T40">ZFMISC_1:40</a></i>;<br/><a NAME="E4:116_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default"><font color="Maroon">b</font></span><a href="borsuk_5.html#NK2" target="_self">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default"><font color="Maroon">b</font></span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</a></span>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="borsuk_5.html#NK2" target="_self">].-infty,</a><span class="default"><font color="Maroon">b</font></span><a href="borsuk_5.html#NK2" target="_self">.[</a></span>, <font color="Maroon">C</font> = <span class="p1"><a href="borsuk_5.html#NK4" target="_self">].</a><span class="default"><font color="Maroon">b</font></span><a href="borsuk_5.html#NK4" target="_self">,+infty.[</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="E6:116_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i><a class="ref" href="borsuk_5.html#T17" target="_self">Th17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:116_1_2_1_1"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 </span><b>by </b><i>, , <a class="ref" href="connsp_1.html#T17">CONNSP_1:17</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:116_1_2_1_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:116_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="borsuk_5.html#T38" target="_self">Th38</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:116_1_2_1_1_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:116_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="borsuk_5.html#T38" target="_self">Th38</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:116_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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