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<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>thus </b><a NAME="E1:111"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a>  implies 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="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>,<font color="Olive">b<sub>2</sub></font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a>  &amp; <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">b<sub>2</sub></font> &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Olive">b<sub>2</sub></font> <a href="tsep_1.html#R1" target="_self">misses</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> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:111_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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="E3:111_1"/><i><font color="Green">E63</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>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E4:111_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font> <a href="tsep_1.html#R1" target="_self">misses</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> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T83" target="_self">Th83</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E5:111_1"/>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a>  &amp; <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>5</sub></font> &amp; ( <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font> <a href="tsep_1.html#R1" target="_self">misses</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> ) )
 <b>by </b><i><a class="txt" href="tsep_1.html#E3:111_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="tsep_1.html#E4:111_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tsep_1.html#T88" target="_self">Th88</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being  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><b> such that </b><a NAME="E2:111"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tsep_1.html#R4" target="_self">are_weakly_separated</a> 
 <b>and </b><br/><a NAME="E3:111"/><i><font color="Green">E64</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>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>5</sub></font> )
 <b>and </b><br/><a NAME="E4:111"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font> <a href="tsep_1.html#R1" target="_self">misses</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>6</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</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>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font>, <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="E1:111_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>8</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="Maroon">c<sub>9</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> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:111_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>5</sub></font> or  not <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_2_1_1_1"><b>suppose </b></a><a NAME="E1:111_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:111_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E2:111"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tsep_1.html#T85" target="_self">Th85</a></i>;<br/><a NAME="E3:111_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="connsp_1.html#R1">are_separated</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</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="tsep_1.html#E3:111"><i><font color="Green">E64</font></i></a>, <a class="txt" href="tsep_1.html#E1:111_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tsep_1.html#D8" target="_self">Def8</a>, <a class="ref" href="tsep_1.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:111_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i><a class="ref" href="connsp_1.html#T8">CONNSP_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_2_1_1_2"><b>suppose </b></a><a NAME="E1:111_2_1_1_2"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R1" target="_self">misses</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:111_2_1_1_2"/>
ex <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 <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>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E1:111_2_1_1_2_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font> &amp; 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#K1" target="_self">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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> )
 
<b>by </b><i><a class="txt" href="tsep_1.html#E1:111_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="tsep_1.html#E1:111_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tsep_1.html#D2" target="_self">Def2</a>, <a class="ref" href="tsep_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E2:111_2_1_1_2_1"/>
( <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E2:111"><i><font color="Green">E63</font></i></a>, <a class="txt" href="tsep_1.html#E3:111"><i><font color="Green">E64</font></i></a>, <a class="txt" href="tsep_1.html#E4:111"><i><font color="Green">E65</font></i></a>, <a class="txt" href="tsep_1.html#E1:111_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="tsep_1.html#E1:111_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tsep_1.html#D3" target="_self">Def3</a>, <a class="ref" href="tsep_1.html#D9" target="_self">Def9</a>, <a class="ref" href="tsep_1.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:111_2_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T67" target="_self">Th67</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:111_2"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="connsp_1.html#R1">are_separated</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:111"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#D8" target="_self">Def8</a></i>;<br/>


</div>
