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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="waybel21.html#M1" target="_self">SemilatticeHomomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>thus </b><a NAME="E1:58"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a> implies <font color="Maroon">c<sub>3</sub></font> is <a href="waybel21.html#V1" target="_self">lim_infs-preserving</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<a NAME="E2:58_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V17">infs-preserving</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 
<b>by </b><i><a class="ref" href="waybel21.html#T46" target="_self">Th46</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel21.html#D4" target="_self">WAYBEL21:def 4</a><br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : verum </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : verum </span>}</span>  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">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel21.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E4:58_1"/><i><font color="Green">E37</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E2:58_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E5:58_1"/><i><font color="Green">E38</font></i>: 
( the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>4</sub></font> &amp;  <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">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></font> #) )
 
<b>by </b><i><a class="ref" href="waybel_9.html#D8">WAYBEL_9:def 8</a></i>;<br/>
<a NAME="E6:58_1"/><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> =  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means verum;<br/>
<a NAME="E7:58_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)</span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="lattice3.html#S2">LATTICE3:sch 2</a>(<a class="txt" href="waybel21.html#E6:58_1"><i><font color="Green">E39</font></i></a>);<br/></i>
<a NAME="E8:58_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</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">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E1:58_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>] means <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>] means <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><a NAME="E2:58_1_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>deffunc </b><font color="Maroon">H<sub>6</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><a NAME="E3:58_1_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><a NAME="E4:58_1_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>by </b><i><a class="txt" href="waybel21.html#E5:58_1"><i><font color="Green">E38</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] means <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E5:58_1_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] iff <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="waybel21.html#E1:58_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E6:58_1_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/>( <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="waybel21.html#E5:58_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>7</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>8</sub></font>) = 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>7</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 

<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">H<sub>7</sub></font>(<font color="Olive">b<sub>1</sub></font>)</span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
<b>from </b><i><a class="ref" href="lattice3.html#S2">LATTICE3:sch 2</a>(<a class="txt" href="waybel21.html#E6:58_1"><i><font color="Green">E39</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 

<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
<b>from </b><i><a class="ref" href="fraenkel.html#S6">FRAENKEL:sch 6</a>(<a class="txt" href="waybel21.html#E2:58_1_1_1"><i><font color="Green">E44</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
<b>from </b><i><a class="ref" href="fraenkel.html#S6">FRAENKEL:sch 6</a>(<a class="txt" href="waybel21.html#E3:58_1_1_1"><i><font color="Green">E45</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
<b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>(<a class="txt" href="waybel21.html#E5:58_1_1_1"><i><font color="Green">E47</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
<b>by </b><i><a class="txt" href="waybel21.html#E5:58_1"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
<b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>(<a class="txt" href="waybel21.html#E6:58_1_1_1"><i><font color="Green">E48</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
<b>from </b><i><a class="ref" href="fraenkel.html#S6">FRAENKEL:sch 6</a>(<a class="txt" href="waybel21.html#E4:58_1_1_1"><i><font color="Green">E46</font></i></a>)</i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> </span>}</span> 

;<br/></div><b>end;</b></div>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E1:58_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>8</sub></font>) <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>8</sub></font>)
 ;<br/>

<a NAME="E2:58_1_1_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:58_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>8</sub></font>) as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:58_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E2:58_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><b>hence </b><a NAME="E4:58_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>8</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>8</sub></font>)</span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:58_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</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">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:58_1_1_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel21.html#E7:58_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E4:58_1_1_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>)</span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="waybel21.html#E2:58_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel21.html#E3:58_1_1_3"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel21.html#E5:58_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E6:58_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>) <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>10</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:58_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:58_1_1_3"/>
<font color="Maroon">c<sub>8</sub></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">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel21.html#E4:58_1_1_3"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:58_1_1"/>
<font color="Maroon">c<sub>8</sub></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">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:58_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> </span>}</span> ,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="waybel21.html#E5:58_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<font color="Maroon">c<sub>8</sub></font> = 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>10</sub></font>)</span>)</span>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="waybel21.html#E1:58_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel21.html#E6:58_1_1"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>10</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel21.html#E2:58_1_1"><i><font color="Green">E43</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:58_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel21.html#E7:58_1"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="waybel11.html#D6">WAYBEL11:def 6</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel21.html#E4:58_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
 <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel21.html#E8:58_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel11.html#D6">WAYBEL11:def 6</a></i>
;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:58"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel21.html#D4" target="_self">WAYBEL21:def 4</a><br/>

<a NAME="E3:58"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <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="waybel_0.html#D37">WAYBEL_0:def 37</a><br/>

<b>assume </b><a NAME="E1:58_2"/>
( not <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> ;<br/>
<b>assume </b><a NAME="E3:58_2"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E4:58_2"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p2">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel17.html#T10">WAYBEL17:10</a></i>
<br/>.= 
 <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel21.html#E2:58"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="waybel21.html#T33" target="_self">Th33</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:58"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel21.html#D1" target="_self">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="waybel20.html#T25">WAYBEL20:25</a></i>;<br/><b>thus </b><a NAME="E2:58_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_3_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a><br/>

<b>thus </b><a NAME="E2:58_3_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="waybel11.html#K1">lim_inf</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel21.html#T28" target="_self">Th28</a></i>
<br/>.= 
 <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel21.html#E2:58"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="waybel21.html#T29" target="_self">Th29</a></i>
<br/>.= 
 <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span>
<b>by </b><i><a class="ref" href="waybel21.html#T28" target="_self">Th28</a></i>
;<br/>
<b>hence </b><a NAME="E4:58_3_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel21.html#T28" target="_self">Th28</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:58"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V17">infs-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel21.html#E4:58"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel_0.html#T71">WAYBEL_0:71</a></i>;<br/>
<b>hence </b><a NAME="E7:58"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="waybel21.html#E3:58"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel21.html#T46" target="_self">Th46</a></i>;<br/>


</div>
