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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</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="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or ( <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>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="borsuk_4.html#E1:49"><i><font color="Green">E40</font></i></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="49_1_1"><b>suppose </b></a><a NAME="E1:49_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:49"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><b>suppose </b></a><a NAME="E1:49_1_2"/><i><font color="Green">E41</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>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>assume </b><a NAME="E2:49_1_2"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:49_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="zfmisc_1.html#T40">ZFMISC_1:40</a></i>;<br/><a NAME="E4:49_1_2"/><i><font color="Green">E43</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E5:49_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
( 0 <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> 1 )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:49_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:49_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:49_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="borsuk_4.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E7:49_1_2"/><i><font color="Green">E46</font></i>: 
( <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="rcomp_2.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>3</sub></font>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_2.html#T17">RCOMP_2:17</a></i>;<br/><a NAME="E8:49_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i><a class="txt" href="borsuk_4.html#E5:49_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E9:49_1_2"/><b>then </b>
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_2.html#K1">.[</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>3</sub></font>,1</span><a href="rcomp_2.html#K2">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_4.html#E1:49_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="borsuk_4.html#E4:49_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="borsuk_4.html#E7:49_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:49_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i><a class="txt" href="borsuk_4.html#E5:49_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="borsuk_4.html#T33" target="_self">Th33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_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="E1:49_1_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="borsuk_4.html#E6:49_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="borsuk_4.html#E11:49_1_2"><i><font color="Green">E47</font></i></a>, <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="49_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:49_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_4.html#E1:49_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:49_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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