<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:109"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <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> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:109"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="tsep_1.html#E1:109"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tsep_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>assume </b><a NAME="E5:109"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 ;<br/><a NAME="E2:109_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D9" target="_self">Def9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E2:109_1_1"/><b>then </b>
( not <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T4" target="_self">Th4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:109_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E5:109_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</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>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> ) )
 <b>by </b><i><a class="txt" href="tsep_1.html#E4:109"><i><font color="Green">E64</font></i></a>, <a class="txt" href="tsep_1.html#E2:109_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tsep_1.html#T66" target="_self">Th66</a></i>;<br/><b>thus </b><a NAME="E6:109_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</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#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>2</sub></font> or ex <font color="Olive">b<sub>3</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:109_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="tsep_1.html#E4:109_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="tsep_1.html#T20" target="_self">Th20</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:109_1_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="tsep_1.html#E4:109_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="tsep_1.html#T20" target="_self">Th20</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:109_1_1_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:109"><i><font color="Green">E63</font></i></a>, <a class="txt" href="tsep_1.html#E4:109"><i><font color="Green">E64</font></i></a>, <a class="txt" href="tsep_1.html#E5:109_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="tsep_1.html#E2:109_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="tsep_1.html#E4:109_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tsep_1.html#D2" target="_self">Def2</a></i>;<br/><b>thus </b><a NAME="E4:109_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:109_1_1_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="tsep_1.html#E3:109_1_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="tsep_1.html#T15" target="_self">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<i><font color="Green">E74</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="tsep_1.html#E4:109"><i><font color="Green">E64</font></i></a>, <a class="txt" href="tsep_1.html#E2:109_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="tsep_1.html#E4:109_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="tsep_1.html#E3:109_1_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="tsep_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tsep_1.html#E2:109_1_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tsep_1.html#D2" target="_self">Def2</a></i>
;<br/>
<a NAME="E4:109_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tsep_1.html#E5:109"><i><font color="Green">E65</font></i></a>, <a class="txt" href="tsep_1.html#E3:109_1_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="tsep_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E5:109_1_1_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:109"><i><font color="Green">E63</font></i></a>, <a class="txt" href="tsep_1.html#E2:109_1_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="tsep_1.html#E3:109_1_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="tsep_1.html#T4" target="_self">Th4</a>, <a class="ref" href="tsep_1.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:109_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>9</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="tsep_1.html#E4:109_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="tsep_1.html#E2:109_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="tsep_1.html#E4:109_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tsep_1.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:109_1"/>
( <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</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#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>2</sub></font> or ex <font color="Olive">b<sub>3</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font> ) ) ) )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</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#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>2</sub></font> or ex <font color="Olive">b<sub>3</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font> ) ) ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:109_2_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E4:109_2_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>7</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:109_2"><i><font color="Green">E67</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E7:109_2_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E8:109_2_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</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="tsep_1.html#E3:109_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="tsep_1.html#T4" target="_self">Th4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_2_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:109_2_1_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:109_2_1_1_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="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K4">\/</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>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
;<br/>

<b>thus </b><a NAME="E1:109_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:109_2_1_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:109_2_1_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:109_2_1_1_1_2"/>
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="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K4">\/</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>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>10</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:109_2_1_1_1_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E4:109"><i><font color="Green">E64</font></i></a>, <a class="txt" href="tsep_1.html#E4:109_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="tsep_1.html#E1:109_2_1_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="tsep_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:109_2_1_1_1_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:109_2_1_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="tsep_1.html#E5:109"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tsep_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E6:109_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="tsep_1.html#E2:109_2_1_1_1_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tsep_1.html#T4" target="_self">Th4</a></i>;<br/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="tsep_1.html#E4:109"><i><font color="Green">E64</font></i></a>, <a class="txt" href="tsep_1.html#E2:109_2_1_1_1_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tsep_1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="tsep_1.html#D2" target="_self">Def2</a></i>
;<br/>
<b>hence </b><a NAME="E8:109_2_1_1_1_2_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="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K4">\/</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>4</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E4:109_2_1_1_1_2_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="tsep_1.html#E6:109_2_1_1_1_2_1"><i><font color="Green">E75</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E10:109_2_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font>, <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>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E4:109"><i><font color="Green">E64</font></i></a>, <a class="txt" href="tsep_1.html#E7:109_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="tsep_1.html#E8:109_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tsep_1.html#T65" target="_self">Th65</a></i>;<br/><b>hence </b><a NAME="E11:109_2_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D9" target="_self">Def9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:109_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T86" target="_self">Th86</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:109"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a>  iff ( <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> or ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1" target="_self">open</a>  <a href="pre_topc.html#M1">SubSpace</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#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>2</sub></font> or ex <font color="Olive">b<sub>3</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>3</sub></font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="tsep_1.html#E6:109"><i><font color="Green">E66</font></i></a></i>;<br/>


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