<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>, <font color="Maroon">T</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">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</font>;<br/>





<b>set </b><font color="Maroon">x</font> =  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font>;<br/>
<b>set </b><font color="Maroon">t</font> =  <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:46"/>
not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>
 ;<br/>

<a NAME="E2:46"/><b>then </b>
not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E3:46"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/>
<a NAME="E4:46"/>
 <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:46"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <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/>
<b>set </b><font color="Maroon">U1</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>;<br/>
<a NAME="E6:46"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">S</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : verum </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">S</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:46_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">S</font>
 ;<br/><b>thus </b><a NAME="E4:46_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">S</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : verum </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="waybel11.html#T30">WAYBEL11:30</a></i>;<br/>
<a NAME="E10:46"/>
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E11:46"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
;<br/>
<a NAME="E12:46"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="waybel11.html#V3">property(S)</a>
 
<b>by </b><i>, <a class="ref" href="waybel11.html#T15">WAYBEL11:15</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E14:46"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="waybel11.html#D3">WAYBEL11:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E16:46"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">S</font>
 <b>by </b><i/>;<br/>
<a NAME="E17:46"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 
;<br/>
<a NAME="E18:46"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E19:46"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> #)
 
<b>by </b><i><a class="ref" href="waybel_9.html#D8">WAYBEL_9:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> ;<br/>
<a NAME="E21:46"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">fy</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j'</font> </span>}</span> ,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">fD</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j1</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">j1</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : verum </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j1</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">j1</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span><b> such that </b><br/><a NAME="E3:46_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">T</font>
 ;<br/><b>thus </b><a NAME="E4:46_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">fD</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j1</font> </span>}</span> ,<font color="Maroon">T</font></span>)</span> where <font color="Olive">j1</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : verum </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E24:46"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j'</font> </span>}</span> ,<font color="Maroon">T</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E25:46"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j'</font> </span>}</span> ,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fD</font>
 
;<br/>
<a NAME="E26:46"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j'</font> </span>}</span> ,<font color="Maroon">T</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">fD</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E27:46"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j'</font> </span>}</span> ,<font color="Maroon">T</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>
 
;<br/>
<a NAME="E28:46"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E29:46"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E30:46"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T138">FUNCT_1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tops_2.html#T52">TOPS_2:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 

;<br/>
<a NAME="E32:46"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p4">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p5">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p0">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="waybel17.html#D1" target="_self">Def1</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E33:46"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p3">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p4">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">N</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>hence </b><a NAME="E34:46"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>


</div>
