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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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="E1:41"/><i><font color="Green">E12</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>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</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>2</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> #)
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="yellow_8.html#V2">irreducible</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="yellow_8.html#R1">is_dense_point_of</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="yellow_8.html#R1">is_dense_point_of</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D6">YELLOW_8:def 6</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="yellow_8.html#V2">irreducible</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D6">YELLOW_8:def 6</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="yellow_8.html#V2">irreducible</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow14.html#E1:41"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow14.html#T24" target="_self">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:41"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="yellow_8.html#R1">is_dense_point_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:41"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="yellow_8.html#R1">is_dense_point_of</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="yellow14.html#E2:41"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="yellow14.html#E1:41"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E8:41"/>
<font color="Maroon">c<sub>6</sub></font> <a href="yellow_8.html#R1">is_dense_point_of</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow14.html#E1:41"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow14.html#E5:41"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow14.html#T25" target="_self">Th25</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow14.html#E1:41"><i><font color="Green">E12</font></i></a></i>;<br/>
<b>assume </b><a NAME="E10:41"/>
<font color="Maroon">c<sub>7</sub></font> <a href="yellow_8.html#R1">is_dense_point_of</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E11:41"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="yellow_8.html#R1">is_dense_point_of</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="yellow14.html#E1:41"><i><font color="Green">E12</font></i></a>, <a class="ref" href="yellow14.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E12:41"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow14.html#E6:41"><i><font color="Green">E15</font></i></a></i>;<br/>


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