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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:37"/><i><font color="Green">E27</font></i>: 
 <a href="waybel_2.html#K5">sup_op</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel14.html#V1" target="_self">jointly_Scott-continuous</a>
 ;<br/>

<a NAME="E2:37"/><i><font color="Green">E28</font></i>: 
 <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#D12">WAYBEL11:def 12</a></i>;<br/>
<a NAME="E3:37"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E4:37"/><i><font color="Green">E30</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="yellow_6.html#K14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T32">WAYBEL11:32</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>2</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>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D6">YELLOW_8:def 6</a><br/>

<a NAME="E5:37"/><i><font color="Green">E31</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="yellow_8.html#D4">YELLOW_8:def 4</a></i>;<br/>
<a NAME="E6:37"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E7:37"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel14.html#T2" target="_self">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E2:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel14.html#E3:37"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel14.html#E4:37"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel14.html#E6:37"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E9:37"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_6.html#V5">prime</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel14.html#E4:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel14.html#T17" target="_self">Th17</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:37"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="waybel14.html#E1:37"><i><font color="Green">E27</font></i></a>, <a class="txt" href="waybel14.html#E7:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel14.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E12:37"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E11:37"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tops_1.html#T21">TOPS_1:21</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E13:37"/><i><font color="Green">E36</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T9">WAYBEL11:9</a></i>;<br/>
<b>hence </b><a NAME="E14:37"/>
<font color="Maroon">c<sub>4</sub></font> <a href="yellow_8.html#R1">is_dense_point_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E12:37"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_8.html#T27">YELLOW_8:27</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E15:37"/>
<font color="Maroon">c<sub>5</sub></font> <a href="yellow_8.html#R1">is_dense_point_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E16:37"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E5:37"><i><font color="Green">E31</font></i></a>, <a class="ref" href="yellow_8.html#T25">YELLOW_8:25</a></i>;<br/>
<a NAME="E17:37"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="t_0topsp.html#V2">T_0</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T10">WAYBEL11:10</a></i>;<br/>
<b>hence </b><a NAME="E18:37"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E12:37"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel14.html#E13:37"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel14.html#E16:37"><i><font color="Green">E37</font></i></a>, <a class="ref" href="yellow_8.html#T32">YELLOW_8:32</a></i>;<br/>


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