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

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font>;<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>7</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E16</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>7</sub></font> #)
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="connsp_1.html#V2">connected</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:10_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E2:10_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="connsp_1.html#R1">are_separated</a> 
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="topalg_5.html#E1:10"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E4:10_1"/>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:10"><i><font color="Green">E16</font></i></a>, <a class="txt" href="topalg_5.html#E2:10_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="topalg_5.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E5:10_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:10"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topalg_5.html#E3:10"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topalg_5.html#E1:10_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="connsp_1.html#T16">CONNSP_1:16</a></i>;<br/><b>hence </b><a NAME="E6:10_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:10"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T16">CONNSP_1:16</a></i>;<br/>


</div>
