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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:98"/><i><font color="Green">E41</font></i>: 
 <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">L</font> is <a href="yellow_6.html#V7">(ITERATED_LIMITS)</a>
 ;<br/>

<b>set </b><font color="Maroon">C</font> =  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">L</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:98_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">I</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">L</font>
 ;<br/><b>let </b><font color="Maroon">K</font> be  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>assume </b><a NAME="E2:98_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</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">L</font>
 ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E3:98_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds   <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Olive">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 ;<br/><b>set </b><font color="Maroon">x</font> =  <a href="yellow_2.html#NK7" target="_self">Inf</a> ;<br/><a NAME="E4:98_1"/><i><font color="Green">E64</font></i>: 
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i><a class="ref" href="waybel_5.html#T16">WAYBEL_5:16</a></i>;<br/><b>reconsider </b><font color="Maroon">r</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/><a NAME="E6:98_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">f</font> =  <a href="waybel_5.html#NK6" target="_self">Sups</a>  as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/><b>set </b><font color="Maroon">X</font> =  <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #);<br/><a NAME="E8:98_1"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #) holds  <font color="Olive">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #), <font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #);<br/>

<a NAME="E1:98_1_1"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #)
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E2:98_1_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:98_1"/><i><font color="Green">E89</font></i>: 
 <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #) is <a href="orders_2.html#V3">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #), <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a><br/>

<b>assume </b><a NAME="E1:98_1_2"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/>

<b>thus </b><a NAME="E2:98_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E10:98_1"/>
 <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #) is <a href="waybel_0.html#V7">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a><br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E1:98_1_3"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">X</font> =  <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">I</font>,<font color="Maroon">r</font>,<font color="Maroon">f</font> #) as  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>) <b>-&gt; </b> non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V6">strict</a>  <a href="waybel_0.html#L1">NetStr</a> of <font color="Maroon">L</font> =  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">J</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E13:98_1"/><i><font color="Green">E90</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i><a NAME="E14:98_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> holds <br/><font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:98_1_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<b>reconsider </b><font color="Maroon">rFi</font> =  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E4:98_1_4"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">rFi</font> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:98_1_4"/>
<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E6:98_1_4"/>
<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel11.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">J</font> = <font color="Maroon">J</font> as    <a href="yellow_6.html#M3">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_6.html#T33">YELLOW_6:33</a></i>;<br/><a NAME="E16:98_1"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> &amp; <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:98_1_5"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<b>take </b><font color="Maroon">R</font> =  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>;<br/>

<b>thus </b><a NAME="E3:98_1_5"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>;<br/>

<b>thus </b><a NAME="E4:98_1_5"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><i><font color="Green">E94</font></i>:  <a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font> = 
<font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="yellow_7.html#T45">YELLOW_7:45</a></i>
<br/>.= 
 <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="waybel11.html#T6" target="_self">Th6</a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>
;<br/><a NAME="E18:98_1"/><i><font color="Green">E95</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E19:98_1"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E20:98_1"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<a NAME="E2:98_1_7"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Ji</font> = <font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font> as  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E4:98_1_7"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</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">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:98_1_7"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Ji</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">L</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T7" target="_self">Th7</a>, </i>;<br/>
<b>hence </b><a NAME="E6:98_1_7"/>
<font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:98_1"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span></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">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<a NAME="E2:98_1_8"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Ji</font> = <font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font> as  <a href="orders_2.html#V2">reflexive</a> <a href="waybel_0.html#V8">monotone</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E4:98_1_8"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:98_1_8"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
;<br/>
<a NAME="E6:98_1_8"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> = 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="waybel11.html#T9" target="_self">Th9</a>, <a class="ref" href="waybel_5.html#D1">WAYBEL_5:def 1</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="waybel11.html#T8" target="_self">Th8</a>, </i>
<br/>.= 
 <a href="waybel_2.html#K1">sup</a> <font color="Maroon">Ji</font>
<b>by </b><i><a class="ref" href="waybel_2.html#D1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">Ji</font>
<b>by </b><i/>
;<br/>
<a NAME="E8:98_1_8"/><b>then </b>
<font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:98_1_8"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font></span>)</span></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">L</font>
 <b>by </b><i><a class="ref" href="waybel11.html#T8" target="_self">Th8</a>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E22:98_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/><a NAME="E23:98_1"/><b>then </b><i><font color="Green">E104</font></i>: 
 <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel11.html#D4" target="_self">Def4</a>, <a class="ref" href="yellow_6.html#T34">YELLOW_6:34</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> : <font color="Olive">p</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/><a NAME="E24:98_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#T35">YELLOW_6:35</a></i>;<br/><b>then reconsider </b><font color="Maroon">G</font> = the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel11.html#D1" target="_self">Def1</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>,   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span>) <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">G</font> <a href="binop_1.html#K2">.</a> <font color="Olive">i</font>,<font color="Maroon">a<sub>2</sub></font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</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> ;<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>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</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">L</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">L</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>,   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</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">L</font> =  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>),<font color="Maroon">L</font>;<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> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">j</font>),<font color="Maroon">L</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> : verum </span>}</span> ;<br/><b>set </b><font color="Maroon">D'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">i</font>,<font color="Olive">g</font>) where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span> ;<br/><b>set </b><font color="Maroon">E'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">g</font>) where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span> ;<br/><a NAME="E26:98_1"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><br/> for <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> holds  <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">g</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">i</font>,<font color="Olive">g</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span>;<br/>



<a NAME="E1:98_1_9"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">j</font>,<font color="Maroon">g</font>) iff  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:98_1_9_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:98_1_9_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_1_9_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">j</font>,<font color="Maroon">g</font>)
 ;<br/><b>then consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:98_1_9_1_1"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">i</font>,<font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:98_1_9_1_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 ;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">i</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><a NAME="E7:98_1_9_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E8:98_1_9_1_1"/><b>then </b><i><font color="Green">E231</font></i>: 
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E9:98_1_9_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel11.html#D2" target="_self">Def2</a>, <a class="ref" href="waybel_5.html#T8">WAYBEL_5:8</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel11.html#D2" target="_self">Def2</a>, <a class="ref" href="waybel11.html#T11" target="_self">Th11</a>, <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="yellow_6.html#D16">YELLOW_6:def 16</a></i>
;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E4:98_1_9_1"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
 <b>and </b><br/><a NAME="E5:98_1_9_1"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E6:98_1_9_1"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D2" target="_self">Def2</a>, , <a class="ref" href="waybel_5.html#T8">WAYBEL_5:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E9:98_1_9_1"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i/>;<br/>
<font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel11.html#D2" target="_self">Def2</a>, <a class="ref" href="waybel11.html#T11" target="_self">Th11</a>, <a class="ref" href="waybel11.html#T12" target="_self">Th12</a>, <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">i</font>,<font color="Maroon">g</font>
<b>by </b><i>, <a class="ref" href="yellow_6.html#D16">YELLOW_6:def 16</a></i>
;<br/>
<b>hence </b><a NAME="E11:98_1_9_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">j</font>,<font color="Maroon">g</font>)
 <b>by </b><i><a class="ref" href="waybel11.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:98_1_9"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">j</font>,<font color="Maroon">g</font>),<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:98_1"/><i><font color="Green">E242</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">j</font>),<font color="Maroon">L</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">i</font>,<font color="Olive">g</font>) where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_10"><b>proof </b></a><div class="add">

<a NAME="E1:98_1_10"/><i><font color="Green">E243</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T35">YELLOW_6:35</a></i>;<br/>
<a NAME="E2:98_1_10"/><i><font color="Green">E244</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span><br/> for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><br/> for <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">g</font></span><a href="domain_1.html#K1">]</a></span> holds <br/> <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">p</font>),<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">i</font>,<font color="Olive">g</font>),<font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:98_1_10_1"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<a NAME="E2:98_1_10_1"/><i><font color="Green">E246</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</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 <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> #)
 
<b>by </b><i><a class="ref" href="yellow_6.html#D16">YELLOW_6:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i><a class="ref" href="waybel11.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E4:98_1_10_1"/>
<font color="Maroon">g'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span>
 
;<br/>
<a NAME="E5:98_1_10_1"/><b>then </b><i><font color="Green">E253</font></i>: 
<font color="Maroon">g'</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E7:98_1_10_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:98_1_10_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">ii</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">ii</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<a NAME="E4:98_1_10_1_1"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>;<br/>
<a NAME="E5:98_1_10_1_1"/>
<font color="Maroon">g'</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">ii</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E8:98_1"><i><font color="Green">E88</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span></span>)</span> ;<br/>
<b>take </b>
 <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<a NAME="E7:98_1_10_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
;<br/>
<b>hence </b><a NAME="E8:98_1_10_1_1"/>
(  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">i'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:98_1_10_1"/><b>then </b><i><font color="Green">E257</font></i>: 
<font color="Maroon">g'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g'</font>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E9:98_1_10_1"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">g</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">p</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_10_1_2"><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="E1:98_1_10_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">g</font>)
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:98_1_10_1_2"/><i><font color="Green">E258</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:98_1_10_1_2"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</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">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i><a class="ref" href="waybel11.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E7:98_1_10_1_2"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">j'</font>,<font color="Maroon">g'</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">g'</font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E8:98_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E8:98_1_10_1_2"/><b>then </b><i><font color="Green">E260</font></i>: 
<font color="Maroon">q</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T12" target="_self">Th12</a>, <a class="ref" href="waybel11.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E9:98_1_10_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E9:98_1"><i><font color="Green">E89</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:98_1_10_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">p</font>)
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E10:98_1_10_1"/><b>then </b><i><font color="Green">E261</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">p</font>),<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">g</font>),<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T3">WAYBEL_7:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>6</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</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">L</font> = <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">g</font>;<br/>
<a NAME="E11:98_1_10_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">g</font>) as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</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"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>7</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</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">L</font> = <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E13:98_1_10_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>7</sub></font>(<font color="Olive">z</font>) where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">z</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">p</font>) as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E15:98_1_10_1"/>
<font color="Maroon">B</font> <a href="yellow_4.html#R4">is_coarser_than</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_10_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_4.html#D2">YELLOW_4:def 2</a><br/>

<b>assume </b><a NAME="E1:98_1_10_1_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span><b> such that </b><br/><a NAME="E3:98_1_10_1_3"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E4:98_1_10_1_3"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">q</font>
 ;<br/>
<b>consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span><b> such that </b><br/><a NAME="E6:98_1_10_1_3"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/>
<b>set </b><font color="Maroon">a</font> = the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">g</font>;<br/>
<a NAME="E8:98_1_10_1_3"/>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">k</font>,<font color="Maroon">g</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E11:98_1_10_1_3"/><i><font color="Green">E267</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">k'</font>,<font color="Maroon">f'</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">g'</font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T12" target="_self">Th12</a>, <a class="ref" href="waybel11.html#T13" target="_self">Th13</a>, , <a class="txt" href="waybel11.html#E22"><i><font color="Green">Lemma92</font></i></a>, </i>;<br/>
<a NAME="E12:98_1_10_1_3"/><b>then </b>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<b>hence </b><a NAME="E13:98_1_10_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<b>set </b><font color="Maroon">N</font> =  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">k'</font></span>)</span>;<br/>
<a NAME="E15:98_1_10_1_3"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">J</font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">k'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g'k</font> = <font color="Maroon">g'</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">k</font>, <font color="Maroon">f'k</font> = <font color="Maroon">f'</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">k'</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel11.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E17:98_1_10_1_3"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">k'</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">f'k</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="waybel11.html#E22"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="yellow_6.html#T36">YELLOW_6:36</a>, <a class="txt" href="waybel11.html#E17:98_1"><i><font color="Green">E94</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">kg</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i><a class="ref" href="waybel11.html#D1" target="_self">Def1</a>, </i>;<br/>
<i><font color="Green">E273</font></i>: <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">kg</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">k'</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">g'k</font>
<b>by </b><i><a class="ref" href="yellow_6.html#T36">YELLOW_6:36</a>, <a class="txt" href="waybel11.html#E17:98_1"><i><font color="Green">E94</font></i></a></i>
;<br/>
<a NAME="E20:98_1_10_1_3"/>
<font color="Maroon">g'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f'</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E16:98_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E21:98_1_10_1_3"/><b>then </b>
<font color="Maroon">g'</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">k</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f'</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>
<b>hence </b><a NAME="E22:98_1_10_1_3"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E17:98_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="waybel11.html#E18:98_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="waybel11.html#E19:98_1"><i><font color="Green">E96</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:98_1_10_1"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">g</font>),<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">p</font>),<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:98_1_10_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">p</font>),<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">g</font>),<font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E9:98_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:98_1_10"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">j</font>),<font color="Maroon">L</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">i</font>,<font color="Olive">g</font>) where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </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="98_1_10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:98_1_10_2"/>
<font color="Maroon">e</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> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">j</font>),<font color="Maroon">L</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span><b> such that </b><br/><a NAME="E3:98_1_10_2"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">p</font>),<font color="Maroon">L</font>
 ;<br/>
<b>consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E5:98_1_10_2"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="waybel11.html#D1" target="_self">Def1</a>, , <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E6:98_1_10_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">j</font>,<font color="Maroon">g</font>),<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T11" target="_self">Th11</a>, <a class="ref" href="waybel11.html#T12" target="_self">Th12</a>, <a class="ref" href="waybel11.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E7:98_1_10_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">i</font>,<font color="Olive">g</font>) where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon">e</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="E4:98_1_10"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">i</font>,<font color="Olive">g</font>) where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E6:98_1_10"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">g</font>),<font color="Maroon">L</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i><a class="ref" href="waybel11.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E8:98_1_10"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">j</font>),<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T11" target="_self">Th11</a>, <a class="ref" href="waybel11.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E9:98_1_10"/>
<font color="Maroon">e</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> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">j</font>),<font color="Maroon">L</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E28:98_1"/><i><font color="Green">E277</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">g</font>) where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">i</font>,<font color="Olive">g</font>) where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="waybel11.html#S1" target="_self">WAYBEL11:sch 1</a>(<a class="ref" href="waybel11.html#T9" target="_self">Th9</a>);<br/></i><a NAME="E29:98_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">g</font>) where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span>  iff  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:98_1_11"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">g</font>) where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span>  implies  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:98_1_11_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">g</font>) where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E3:98_1_11_1"/><i><font color="Green">E278</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"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
 ;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E4:98_1_11_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel11.html#D3" target="_self">Def3</a></i>;<br/>

<a NAME="E5:98_1_11_1"/><b>then </b><i><font color="Green">E279</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span>
 
;<br/>
<b>thus </b><font color="Maroon">y</font> = 
 <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="waybel11.html#T11" target="_self">Th11</a>, <a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="waybel11.html#T12" target="_self">Th12</a>, <a class="ref" href="waybel_5.html#D2">WAYBEL_5:def 2</a></i>
;<br/>


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

<b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:98_1_11"/><i><font color="Green">E280</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 <b>and </b><br/><a NAME="E3:98_1_11"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i><a class="ref" href="waybel11.html#D2" target="_self">Def2</a>, <a class="ref" href="waybel11.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:98_1_11"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T11" target="_self">Th11</a></i>;<br/>
<b>then </b><font color="Maroon">y</font> = 
 <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="waybel11.html#T12" target="_self">Th12</a>, <a class="ref" href="waybel_5.html#D2">WAYBEL_5:def 2</a></i>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E7:98_1_11"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">g</font>) where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E30:98_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">g</font>) where <font color="Olive">g</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E31:98_1"/><b>then </b><i><font color="Green">E282</font></i>: 
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> =  <a href="yellow_2.html#NK7" target="_self">Inf</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><b>set </b><font color="Maroon">X1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where <font color="Olive">j'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> ;<br/><b>set </b><font color="Maroon">X2</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">X</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">X</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> ;<br/><a NAME="E33:98_1"/><i><font color="Green">E283</font></i>: 
<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">X</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">X</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where <font color="Olive">j'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_12"><b>proof </b></a><div class="add">

<a NAME="E1:98_1_12"/><i><font color="Green">E284</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds   <a href="yellow_2.html#NK7" target="_self">Inf</a>  <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">X</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">X</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>set </b><font color="Maroon">X3</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">X</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">X</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ;<br/>
<a NAME="E1:98_1_12_1"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">X</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">X</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span>  iff  ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> &amp; <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_12_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_1_12_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">X</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">X</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>then consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:98_1_12_1_1_1"/><i><font color="Green">E285</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E4:98_1_12_1_1_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 ;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">i</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">u</font>;<br/><a NAME="E6:98_1_12_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>hence </b><a NAME="E7:98_1_12_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>thus </b><a NAME="E8:98_1_12_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:98_1_12_1_1"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>
 <b>and </b><br/><a NAME="E3:98_1_12_1_1"/><i><font color="Green">E287</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">u</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:98_1_12_1_1"/><i><font color="Green">E288</font></i>: 
<font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:98_1_12_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E8:98_1"><i><font color="Green">E88</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:98_1_12_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">X</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">X</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#E9:98_1"><i><font color="Green">E89</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:98_1_12_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">X</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">X</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="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:98_1_12_1"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <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">X</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">X</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">L</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:98_1_12"/>
<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">X</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">X</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where <font color="Olive">j'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : 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="98_1_12_2"><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="E1:98_1_12_2"/>
<font color="Maroon">u</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">X</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">X</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</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">X</font><b> such that </b><br/><a NAME="E3:98_1_12_2"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">u</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">X</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">X</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">L</font>
 ;<br/>
<a NAME="E4:98_1_12_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<b>by </b><i><a class="ref" href="waybel11.html#T13" target="_self">Th13</a>, </i>;<br/>
<b>hence </b><a NAME="E5:98_1_12_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where <font color="Olive">j'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/>


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

<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>consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>
<b>assume </b><a NAME="E4:98_1_12"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where <font color="Olive">j'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E5:98_1_12"/><b>then </b>
 ex <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> st <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
;<br/>
<a NAME="E6:98_1_12"/><b>then </b>
<font color="Maroon">u</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">X</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">X</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">L</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E7:98_1_12"/>
<font color="Maroon">u</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">X</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">X</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:98_1_13"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where <font color="Olive">j'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  iff  ex <font color="Olive">j'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> st <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  )
 ;<br/><a NAME="E2:98_1_13"/><b>then </b>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where <font color="Olive">j'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  iff <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  )
 ;<br/><b>hence </b><a NAME="E3:98_1_13"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where <font color="Olive">j'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span>  iff <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>then </b> <a href="yellow_0.html#K1">"\/"</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">X</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">X</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> ,<font color="Maroon">L</font> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x'</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="waybel11.html#T12" target="_self">Th12</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>
<br/>.= 
 <a href="yellow_2.html#NK7" target="_self">Inf</a> 
<b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>
;<br/><a NAME="E36:98_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E37:98_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E38:98_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span></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">L</font>
 <b>by </b><i><a class="ref" href="waybel11.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E39:98_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span></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">L</font>
 <b>by </b><i><a class="ref" href="waybel11.html#T4" target="_self">Th4</a>, <a class="ref" href="waybel11.html#T7" target="_self">Th7</a>, <a class="ref" href="yellow_6.html#D26">YELLOW_6:def 26</a></i>;<br/><a NAME="E40:98_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="waybel11.html#R1" target="_self">is_S-limit_of</a>  <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">J</font>
 <b>by </b><i>, </i>;<br/><a NAME="E41:98_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i><a class="ref" href="waybel11.html#T11" target="_self">Th11</a>, </i>;<br/><b>hence </b><a NAME="E42:98_1"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:98"/>
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="ref" href="waybel_5.html#T18">WAYBEL_5:18</a></i>;<br/>


</div>
