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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">V</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E2:34"/><i><font color="Green" title="E25">A2</font></i>: 
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#D12" title="WAYBEL11:def.12">WAYBEL11:def 12</a></i>;<br/>
<a NAME="E3:34"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T32" title="WAYBEL11:th.32">WAYBEL11:32</a></i>;<br/>
<b>given </b><font color="Maroon" title="c4">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><a NAME="E5:34"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>

<a NAME="E6:34"/><i><font color="Green" title="E28">A5</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">u</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T9" title="WAYBEL11:th.9">WAYBEL11:9</a></i>;<br/>
<a NAME="E7:34"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">irreducible</a>
 
<b>by </b><i><a class="ref" href="yellow_8.html#T26" title="YELLOW_8:th.26">YELLOW_8:26</a></i>;<br/>
<b>hence </b><a NAME="E8:34"/>
<font color="Maroon" title="c3">V</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="waybel14.html#E2:34"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="waybel14.html#E3:34"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="waybel14.html#E5:34"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="waybel14.html#E6:34"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="waybel14.html#T17" target="_self" title="WAYBEL14:th.17">Th17</a></i>;<br/>

<b>assume </b><a NAME="E9:34"/>
<font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>hence </b><a NAME="E10:34"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E1:34"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="waybel14.html#E5:34"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="waybel14.html#T2" target="_self" title="WAYBEL14:th.2">Th2</a></i>;<br/>


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