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<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>, <font color="Maroon">c<sub>4</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="E1:87"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<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/>
<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>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>assume </b><a NAME="E5:87"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 ;<br/>

<b>thus </b><a NAME="E6:87"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

<a NAME="E1:87_1"/><i><font color="Green">E59</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="tsep_1.html#E5:87"><i><font color="Green">E57</font></i></a>, <a class="ref" href="tsep_1.html#D8" target="_self">Def8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1"><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="E1:87_1_1"/>
( <font color="Maroon">c<sub>8</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>3</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &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 <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>2</sub></font></span>)</span> )
 ;<br/><a NAME="E2:87_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</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="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:87"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tsep_1.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E3:87_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="txt" href="tsep_1.html#E1:87_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="tsep_1.html#T43" target="_self">Th43</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:87_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>2</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><b>end;</b></div>

<b>thus </b><a NAME="E7:87"/>
<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>,<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>proof </b></a><div class="add">

<a NAME="E1:87_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 
<b>by </b><i><a class="txt" href="tsep_1.html#E1:87"><i><font color="Green">E56</font></i></a>, <a class="txt" href="tsep_1.html#E6:87"><i><font color="Green">E58</font></i></a>, <a class="ref" href="tsep_1.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E2:87_2"/>
<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>,<font color="Maroon">c<sub>2</sub></font> <a href="tsep_1.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:87"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tsep_1.html#T29" target="_self">Th29</a></i>;<br/>


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


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