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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:36"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">J</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st <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> holds <br/>for <font color="Olive">K</font> being <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">J</font>  st ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">J</font> holds  <font color="Olive">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> ) holds <br/>for <font color="Olive">F</font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Olive">K</font>,<font color="Maroon">L</font>  st ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">J</font> holds   <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a> ) holds <br/> <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> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>set </b><font color="Maroon">J</font> = <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">j</font> </span>}</span> ;<br/><b>set </b><font color="Maroon">UN</font> =  <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/><a NAME="E1:36_1"/><i><font color="Green">E36</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5">the_transitive-closure_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_6.html#D3">YELLOW_6:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">UN</font> =  <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> as  <a href="classes2.html#V1">universal</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E3:36_1"/><i><font color="Green">E38</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K5">the_transitive-closure_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="classes1.html#T59">CLASSES1:59</a></i>;<br/><a NAME="E4:36_1"/>
 <a href="classes1.html#K5">the_transitive-closure_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UN</font>
 <b>by </b><i>, <a class="ref" href="classes1.html#T5">CLASSES1:5</a></i>;<br/><a NAME="E5:36_1"/><b>then </b><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UN</font>
 <b>by </b><i>, <a class="ref" href="classes1.html#D1">CLASSES1:def 1</a></i>;<br/><a NAME="E6:36_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UN</font>
 <b>by </b><i><a class="ref" href="classes2.html#T65">CLASSES2:65</a></i>;<br/><a NAME="E7:36_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">j</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:36_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">j</font> </span>}</span> 
 ;<br/>

<a NAME="E2:36_1_1"/><b>then </b>
 ex <font color="Olive">j</font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">j</font> )
 
;<br/>
<b>hence </b><a NAME="E3:36_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E8:36_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">j</font> </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">UN</font>
 <b>by </b><i>, <a class="ref" href="classes1.html#D1">CLASSES1:def 1</a></i>;<br/><a NAME="E9:36_1"/><i><font color="Green">E55</font></i>: 
<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> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T5">WAYBEL_0:5</a></i>;<br/><a NAME="E10:36_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <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="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/><a NAME="E11:36_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<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> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">j</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">J</font> = <span class="p1">{<span class="default"> <font color="Olive">j</font> where <font color="Olive">j</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">j</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E13:36_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds  <font color="Olive">j</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>

<a NAME="E1:36_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 
;<br/>
<b>then consider </b><font color="Maroon">j'</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:36_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">j'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E4:36_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">j'</font>
 ;<br/>
<b>thus </b><a NAME="E5:36_1_2"/>
<font color="Maroon">j</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:36"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">K</font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">J</font>;<br/><a NAME="E14:36_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/><b>then reconsider </b><font color="Maroon">K</font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">J</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E16:36_1"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</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> <font color="Maroon">UN</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>

<a NAME="E1:36_1_3"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E2:36_1_3"/><b>then </b>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 
;<br/>
<b>hence </b><a NAME="E3:36_1_3"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UN</font>
 <b>by </b><i>, , <a class="ref" href="classes1.html#D1">CLASSES1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:36_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> <font color="Maroon">J</font> holds <br/> not <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is <a href="xboole_0.html#V1">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_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="E1:36_1_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</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">J</font> ;<br/>
<a NAME="E3:36_1_4"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i'</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E4:36_1_4"/>
not <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">K</font> = <font color="Maroon">K</font> as  <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> =  <a href="partfun1.html#K6">id</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>;<br/><a NAME="E19:36_1"/>
 ex <font color="Olive">F</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp; ( 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">J</font> holds <br/><font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><b>then consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E21:36_1"/><i><font color="Green">E80</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 <b>and </b><br/><a NAME="E22:36_1"/><i><font color="Green">E81</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">J</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E24:36_1"/>
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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> is   <a href="funct_1.html#NM1">Function</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:36_1_5"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:36_1_5"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="waybel_5.html#E1:36_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:36_1_5"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/><a NAME="E26:36_1"/>
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">J</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:36_1_6"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> ;<br/>
<a NAME="E3:36_1_6"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:36_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E4:36_1_6"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:36_1_6"/><i><font color="Green">E98</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E6:36_1_6"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E7:36_1_6"/><b>then </b>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:36_1_6"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E5:36_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:36_1_6"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E10:36_1_6"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E3:36_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E28:36_1"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> holds  <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>;<br/>



<a NAME="E1:36_1_7"/>
<font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:36_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:36_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>


</div><b>end;</b></div><a NAME="E29:36_1"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</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" target="_self">.</a> <font color="Olive">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_8_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <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" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_8_1"/><i><font color="Green">E101</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"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E4:36_1_8_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:36_1_8_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E5:36_1"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E6:36_1_8_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E7:36_1_8_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E2:36_1_8"/><b>then </b><i><font color="Green">E104</font></i>: 
 <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" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_8_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:36_1_8_2"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E3:36_1_8_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a></i>;<br/><a NAME="E4:36_1_8_2"/>
<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"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E5:36_1_8_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <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" target="_self">.</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/></div><b>end;</b></div>
<a NAME="E4:36_1_8"/><b>then </b>
<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a>  <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" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_8"/>
 <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" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E5:36_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:36_1"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</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" target="_self">.</a> <font color="Olive">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>

<a NAME="E1:36_1_9"/>
 <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" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:36_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:36_1_9"/>
 <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" target="_self">.</a> <font color="Maroon">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E31:36_1"/>
for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</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" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> holds <br/> <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" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font></span>)</span>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:36_1_10"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">f</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" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">a</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" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font>;<br/>
<a NAME="E2:36_1_10"/>
for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D</font> holds <br/> ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> &amp;  <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" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">d</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:36_1_10_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:36_1_10_1"/><b>then </b>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> ;<br/>
<a NAME="E4:36_1_10_1"/><i><font color="Green">E114</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>

<a NAME="E1:36_1_10_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 
;<br/>
<a NAME="E2:36_1_10_1_1"/><b>then </b>
<font color="Maroon">j</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/>
<a NAME="E3:36_1_10_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:36_1_10_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<i><font color="Green">E115</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>
<br/>.= 
<font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_10_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_10_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_10_1_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:36_1_10_1_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i><a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="txt" href="waybel_5.html#E6:36_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E6:36_1_10_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="waybel_5.html#E7:36_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E7:36_1_10_1_3"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E8:36_1_10_1_3"/>
<font color="Maroon">j</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:36_1_10_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:36_1_10_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">J</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i><a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E9:36_1_10_1"/><i><font color="Green">E119</font></i>: 
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">D'</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T55">YELLOW_2:55</a></i>;<br/>
<i><font color="Green">E120</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_10_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_10_1_5"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:36_1_10_1_5"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">j</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="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:36_1_10_1_5"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j'</font>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j'</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j'</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, </i>
;<br/></div><b>end;</b></div>
<a NAME="E12:36_1_10_1"/><b>then </b><i><font color="Green">E123</font></i>: 
 <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" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">D'</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E6:36_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_5.html#E7:36_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E13:36_1_10_1"/>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">D'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:36_1_10_1"/><b>then </b>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">D'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E15:36_1_10_1"/>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp;  <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" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">d</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:36_1_10"/><b>then </b>
 <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" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a></i>;<br/>
<a NAME="E4:36_1_10"/><b>then </b>
 <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" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_10"/>
 <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" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:36_1"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E33:36_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a>  <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_11"><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="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:36_1_11"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <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>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><b> such that </b><br/><a NAME="E3:36_1_11"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i/>;<br/>
<a NAME="E4:36_1_11"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 
;<br/>
<b>then consider </b><font color="Maroon">j'</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:36_1_11"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">j'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E7:36_1_11"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">j'</font>
 ;<br/>
<font color="Maroon">b</font> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">j'</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E3:36_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:36_1_11"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E6:36_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E34:36_1"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/><a NAME="E35:36_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 <b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/><b>reconsider </b><font color="Maroon">j</font> = <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> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i><a class="txt" href="waybel_5.html#E1"><i><font color="Green">Lemma32</font></i></a></i>;<br/> <a href="yellow_2.html#NK6" target="_self">Sup</a>  = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>
<br/>.= 
 <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="txt" href="waybel_5.html#E3:36_1"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>
;<br/><a NAME="E38:36_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <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>
 <b>by </b><i/>;<br/><a NAME="E39:36_1"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/><a NAME="E40:36_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/><b>then </b><i><font color="Green">E129</font></i>: <font color="Maroon">x</font> = 
 <a href="yellow_2.html#NK7" target="_self">Inf</a> 
<b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i>, , <a class="txt" href="waybel_5.html#E1:36"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel_5.html#E5:36_1"><i><font color="Green">E39</font></i></a></i>
;<br/><a NAME="E42:36_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T9">WAYBEL_3:9</a></i>;<br/><a NAME="E43:36_1"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><b>hence </b><a NAME="E44:36_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:36"/><b>then </b><i><font color="Green">E130</font></i>: 
( <font color="Maroon">L</font> is <a href="waybel_0.html#V24">up-complete</a> &amp; <font color="Maroon">L</font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E4:36"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="waybel_0.html#V1">directed</a> )
 
;<br/>
<b>hence </b><a NAME="E5:36"/>
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>


</div>
