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<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D13" target="_self">BORSUK_1:def 13</a><br/>

<b>assume </b><a NAME="E1:96_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K16" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="connsp_2.html#M2">a_neighborhood</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>

<a NAME="E2:96_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E3:96_1_1"/><i><font color="Green">E74</font></i>: 
<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>5</sub></font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D2">CONNSP_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="borsuk_1.html#E1:96_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="borsuk_1.html#T74" target="_self">Th74</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:96_1_1_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="borsuk_1.html#E1:96_1_1_1_1_1"><i><font color="Green">E75</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E4:96_1_1_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/><a NAME="E5:96_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:96_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E6:96_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:96_1_1_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/><a NAME="E7:96_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is    <a href="connsp_2.html#M2">a_neighborhood</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D2">CONNSP_2:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:96_1_1_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E10:96_1_1_1_1_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E11:96_1_1_1_1_1"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:96_1_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="borsuk_1.html#D13" target="_self">Def13</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:96_1_1_1_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E14:96_1_1_1_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E9:96_1_1_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/><b>thus </b><a NAME="E15:96_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E13:96_1_1_1_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/><a NAME="E16:96_1_1_1_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E14:96_1_1_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E17:96_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E10:96_1_1_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E18:96_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:96_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E19:96_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E20:96_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:96_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E21:96_1_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E22:96_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E10:96_1_1_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E23:96_1_1_1_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E16:96_1_1_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E24:96_1_1_1_1_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K16" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="borsuk_1.html#E24:96_1_1_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="borsuk_1.html#T74" target="_self">Th74</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:96_1_1_1_1_1_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:96_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E24:96_1_1_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_1.html#T74">XBOOLE_1:74</a></i>;<br/><a NAME="E3:96_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E14:96_1_1_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="borsuk_1.html#E1:96_1_1_1_1_1_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_1.html#T77">XBOOLE_1:77</a></i>;<br/><a NAME="E4:96_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E11:96_1_1_1_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="borsuk_1.html#E1:96_1_1_1_1_1_1_1_1"><i><font color="Green">E85</font></i></a></i>;<br/><b>hence </b><a NAME="E5:96_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E23:96_1_1_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:96_1_1_1_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:96_1_1_1_1_1_1_1_2"/><i><font color="Green">E85</font></i>: 
( not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><a NAME="E4:96_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E24:96_1_1_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="borsuk_1.html#E3:96_1_1_1_1_1_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/><b>hence </b><a NAME="E5:96_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:96_1_1_1_1_1_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E26:96_1_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:96_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:96_1_1_1_1_2"/><i><font color="Green">E75</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:96_1_1_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E5:96_1_1_1_1_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E7:96_1_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E8:96_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/><a NAME="E9:96_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/><b>hence </b><a NAME="E10:96_1_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/><a NAME="E11:96_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:96_1_1_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E12:96_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:96_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="borsuk_1.html#E3:96_1_1_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E13:96_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:96_1_1_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E14:96_1_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><b>hence </b><a NAME="E15:96_1_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:96_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E16:96_1_1_1_1_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K16" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_1_1_1_1_2_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:96_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><b>hence </b><a NAME="E3:96_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_1.html#E16:96_1_1_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="borsuk_1.html#E1:96_1_1_1_1_2_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E19:96_1_1_1_1_2"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E20:96_1_1_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E16:96_1_1_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="borsuk_1.html#T74" target="_self">Th74</a></i>;<br/><a NAME="E21:96_1_1_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E16:96_1_1_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="borsuk_1.html#E20:96_1_1_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/><b>hence </b><a NAME="E22:96_1_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E20:96_1_1_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:96_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K16" target="_self">TrivExt</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>


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