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

<b>set </b><font color="Maroon">S</font> =  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font>;<br/>
<b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D24">YELLOW_6:def 24</a><br/><b>let </b><font color="Maroon">Y</font> be    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font>;<br/>



<a NAME="E1:71_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D21">YELLOW_6:def 21</a></i>;<br/>
<b>assume </b><a NAME="E2:71_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y'</font> being  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:71_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">Y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:71_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y'</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>
<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<b>assume </b><a NAME="E7:71_1_1"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E8:71_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">N'</font> being  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E10:71_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">N'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E11:71_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N'</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</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">N</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">R</font> =  <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">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ,<font color="Maroon">R</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<b>reconsider </b><font color="Maroon">X1</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</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">R</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ,<font color="Maroon">R</font>;<br/>
<b>reconsider </b><font color="Maroon">X2</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">j</font>) where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<a NAME="E14:71_1_1"/>
<font color="Maroon">p</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">N'</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E15:71_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">N</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:71_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">p'</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X1</font>,<font color="Maroon">R</font>
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">N</font><b> such that </b><br/><a NAME="E18:71_1_1"/><i><font color="Green">E90</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Y</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E19:71_1_1"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  ex <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">p</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a></i>;<br/>
<a NAME="E20:71_1_1"/>
<font color="Maroon">X1</font> <a href="yellow_4.html#R3">is_finer_than</a> <font color="Maroon">X2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_4.html#D1">YELLOW_4:def 1</a><br/>

<b>assume </b><a NAME="E1:71_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<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="E3:71_1_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">a</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">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">R</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E6:71_1_1_1"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>  st <font color="Maroon">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Maroon">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">p</font>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">X3</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">X4</font> = <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">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ;<br/>
<b>take </b><font color="Maroon">b</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> </span>}</span> ,<font color="Maroon">R</font>;<br/>

<b>thus </b><a NAME="E7:71_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E8:71_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> </span>}</span>  <a href="tarski.html#R1">c=</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">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</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="E1:71_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:71_1_1_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E4:71_1_1_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">m</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E6:71_1_1_1_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">m</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:71_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E8:71_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</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">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel11.html#E2:71_1_1"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:71_1_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="waybel_7.html#T3">WAYBEL_7:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:71_1_1"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X1</font>,<font color="Maroon">R</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X2</font>,<font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E22:71_1_1"/><b>then </b>
<font color="Maroon">p'</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X2</font>,<font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E23:71_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">Y</font>
 
;<br/>
<a NAME="E24:71_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">Y'</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E25:71_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:71_1_1"><i><font color="Green">E41</font></i></a>, , </i>;<br/>


</div>
