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<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E20</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="E2:15"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="connsp_2.html#V1">locally_connected</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>7</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D4">CONNSP_2:def 4</a><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D3">CONNSP_2:def 3</a><br/>



<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</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#E1:15"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="topalg_5.html#E1:15"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E5:15"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="connsp_2.html#R1">is_locally_connected_in</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:15"><i><font color="Green">E21</font></i></a>, <a class="ref" href="connsp_2.html#D4">CONNSP_2:def 4</a></i>;<br/>
<b>assume </b><a NAME="E6:15"/>
<font color="Maroon">c<sub>9</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E7:15"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:15"><i><font color="Green">E20</font></i></a>, <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E9:15"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E10:15"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>and </b><br/><a NAME="E11:15"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:15"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_2.html#D3">CONNSP_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="topalg_5.html#E1:15"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E13:15"/>
<font color="Maroon">c<sub>13</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:15"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topalg_5.html#E9:15"><i><font color="Green">E23</font></i></a>, <a class="ref" href="topalg_5.html#T8" target="_self">Th8</a></i>;<br/>

<b>thus </b><a NAME="E14:15"/>
<font color="Maroon">c<sub>13</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:15"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topalg_5.html#E10:15"><i><font color="Green">E24</font></i></a>, <a class="ref" href="topalg_5.html#T7" target="_self">Th7</a></i>;<br/>

<b>thus </b><a NAME="E15:15"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:15"><i><font color="Green">E25</font></i></a></i>;<br/>


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