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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>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>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>6</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>7</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>thus </b><a NAME="E4:90"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a>  &amp; <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>  implies <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>,<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="90_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:90_1"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a>  &amp; <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/>

<a NAME="E2:90_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>5</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>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="connsp_1.html#R1">are_separated</a>  )
 
<b>by </b><i><a class="ref" href="tsep_1.html#D8" target="_self">Def8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_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="E1:90_1_1"/><i><font color="Green">E58</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 <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> &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>4</sub></font> )
 ;<br/><a NAME="E2:90_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</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="E3:90_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#E2:90_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="tsep_1.html#E1:90_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="tsep_1.html#T45" target="_self">Th45</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:90_1"/>
<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>,<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="ref" href="tsep_1.html#D8" target="_self">Def8</a></i>;<br/>


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

<b>assume </b><a NAME="E5:90"/><i><font color="Green">E57</font></i>: 
<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>,<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a> 
 ;<br/>

<a NAME="E6:90"/>
( <font color="Maroon">c<sub>2</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#K1" target="_self">union</a> <font color="Maroon">c<sub>3</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>2</sub></font> <a href="tsep_1.html#K1" target="_self">union</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2" target="_self">Th2</a>, <a class="ref" href="tsep_1.html#T22" target="_self">Th22</a></i>;<br/>
<b>hence </b><a NAME="E7:90"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R3" target="_self">are_separated</a>  &amp; <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>  )
 <b>by </b><i><a class="txt" href="tsep_1.html#E5:90"><i><font color="Green">E57</font></i></a>, <a class="ref" href="tsep_1.html#T77" target="_self">Th77</a></i>;<br/>


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