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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="waybel_0.html#V25" title="WAYBEL_0:attr.25">/\-complete</a> <a href="waybel32.html#V2" title="WAYBEL32:attr.2">order_consistent</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/><b>let </b><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>





<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> =  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> $1 </span>}</span> ,<font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span>}</span> ;<br/>
<a NAME="E2:48"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span>}</span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8" title="DOMAIN_1:sch.8">DOMAIN_1:sch 8</a>();<br/></i>
<b>then reconsider </b><font color="Maroon" title="c5">D</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">D</font> = <font color="Maroon" title="c5">D</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="waybel32.html#T11" target="_self" title="WAYBEL32:th.11">Th11</a></i>;<br/>
<a NAME="E5:48"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c2">N</font>
 
;<br/>
<a NAME="E6:48"/><b>then </b><i><font color="Green" title="E24">A2</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel32.html#K2" title="WAYBEL32:func.2">inf_net</a> <font color="Maroon" title="c2">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel32.html#T29" target="_self" title="WAYBEL32:th.29">Th29</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_9.html#D9" title="WAYBEL_9:def.9">WAYBEL_9:def 9</a><br/>

<a NAME="E7:48"/><i><font color="Green" title="E25">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds   <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b1">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel32.html#D2" target="_self" title="WAYBEL32:def.2">Def2</a></i>;<br/>
<a NAME="E8:48"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T51" title="TOPS_1:th.51">TOPS_1:51</a></i>;<br/>
<a NAME="E9:48"/><b>then </b><i><font color="Green" title="E26">A4</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">V</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 
<b>by </b><i><a class="txt" href="waybel32.html#E7:48"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="waybel32.html#T25" target="_self" title="WAYBEL32:th.25">Th25</a></i>;<br/>
<a NAME="E10:48"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E11:48"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">V</font>
 
<b>by </b><i><a class="txt" href="waybel32.html#E1:48"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="waybel32.html#E9:48"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">W</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">V</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">D</font> <b>by </b><i><a class="ref" href="connsp_2.html#T5" title="CONNSP_2:th.5">CONNSP_2:5</a>, <a class="ref" href="tops_1.html#T51" title="TOPS_1:th.51">TOPS_1:51</a></i>;<br/>
<a NAME="E13:48"/><i><font color="Green" title="E27">A5</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<a NAME="E14:48"/>
 <a href="waybel32.html#K2" title="WAYBEL32:func.2">inf_net</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c8">W</font>
 
<b>by </b><i><a class="txt" href="waybel32.html#E6:48"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="waybel32.html#D2" target="_self" title="WAYBEL32:def.2">Def2</a></i>;<br/>
<a NAME="E15:48"/><b>then </b>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c8">W</font>
 
<b>by </b><i><a class="txt" href="waybel32.html#E9:48"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="waybel32.html#T31" target="_self" title="WAYBEL32:th.31">Th31</a></i>;<br/>
<a NAME="E16:48"/><b>then </b>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c7">V</font>
 
<b>by </b><i><a class="txt" href="waybel32.html#E13:48"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="waybel_0.html#T8" title="WAYBEL_0:th.8">WAYBEL_0:8</a></i>;<br/>
<b>hence </b><a NAME="E17:48"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c7">V</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T28" title="YELLOW_6:th.28">YELLOW_6:28</a></i>;<br/>


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