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

<b>given </b><font color="Maroon">c<sub>7</sub></font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><a NAME="E1:20"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="connsp_1.html#V2">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>6</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>6</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>assume </b><a NAME="E2:20"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>9</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a><br/>

<a NAME="E3:20"/><i><font color="Green">E26</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:20"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E6:20"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E7:20"/><i><font color="Green">E28</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:20"><i><font color="Green">E26</font></i></a>, <a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:20"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E10:20"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:20"><i><font color="Green">E25</font></i></a>, <a class="txt" href="topalg_5.html#E7:20"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</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#E10:20"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<a NAME="E12:20"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<a NAME="E13:20"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E6:20"><i><font color="Green">E27</font></i></a>, <a class="txt" href="topalg_5.html#E10:20"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E14:20"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E12:20"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E15:20"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:20"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a><br/>

<b>thus </b><a NAME="E16:20"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:20"><i><font color="Green">E24</font></i></a>, <a class="txt" href="topalg_5.html#E6:20"><i><font color="Green">E27</font></i></a>, <a class="txt" href="topalg_5.html#E10:20"><i><font color="Green">E30</font></i></a></i>;<br/>

<a NAME="E17:20"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E10:20"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E18:20"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E19:20"/>
<font color="Maroon">c<sub>12</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#E7:20"><i><font color="Green">E28</font></i></a>, <a class="txt" href="topalg_5.html#E17:20"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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