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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>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <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>;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:34"/><i><font color="Green">E25</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="E3:34"/><i><font color="Green">E26</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>given </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><a NAME="E4:34"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</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> 
 ;<br/>

<a NAME="E5:34"/><i><font color="Green">E28</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/>
<a NAME="E6:34"/>
 <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> is <a href="yellow_8.html#V2">irreducible</a>
 
<b>by </b><i><a class="ref" href="yellow_8.html#T26">YELLOW_8:26</a></i>;<br/>
<b>hence </b><a NAME="E7:34"/>
<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#E1:34"><i><font color="Green">E24</font></i></a>, <a class="txt" href="waybel14.html#E2:34"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel14.html#E3:34"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel14.html#E4:34"><i><font color="Green">E27</font></i></a>, <a class="txt" href="waybel14.html#E5:34"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel14.html#T17" target="_self">Th17</a></i>;<br/>

<b>assume </b><a NAME="E8:34"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>hence </b><a NAME="E9:34"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E1:34"><i><font color="Green">E24</font></i></a>, <a class="txt" href="waybel14.html#E4:34"><i><font color="Green">E27</font></i></a>, <a class="ref" href="waybel14.html#T2" target="_self">Th2</a></i>;<br/>


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