<?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="E1:66"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/>

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

<b>given </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">S</font><b> such that </b><a NAME="E2:66"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">g</font> is <a href="waybel_0.html#V17">infs-preserving</a> &amp; <font color="Maroon">g</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 <b>and </b><br/><a NAME="E3:66"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">g</font> is <a href="funct_2.html#V2">onto</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">S'</font> = <font color="Maroon">S</font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i>, , <a class="txt" href="waybel_5.html#E3:66"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E5:66"/>
for <font color="Olive">J</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <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><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">S'</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">J</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">K</font> be  <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</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">S'</font>;<br/>





<b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E39</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>
 ;<br/>

<b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:66_1"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="waybel_1.html#K1">]</a></span> is <a href="waybel_1.html#V3">Galois</a>
 <b>and </b><br/><a NAME="E4:66_1"/>
for <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Maroon">d</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">t</font> <a href="waybel_1.html#R3">is_minimum_of</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E3:66_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_1.html#T15">WAYBEL_1:15</a></i>;<br/>
<a NAME="E5:66_1"/>
for <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Maroon">d</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">t</font> <a href="waybel_1.html#R3">is_minimum_of</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">t</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E7:66_1"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="waybel_1.html#T23">WAYBEL_1:23</a></i>;<br/>
<a NAME="E6:66_1"/><b>then </b>
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T24">WAYBEL_1:24</a></i>;<br/>
<a NAME="E7:66_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">d</font> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>;<br/>
<a NAME="E8:66_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">d</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#T9">WAYBEL_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">dF</font> = <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="pboole.html#K12">**</a> <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="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a></i>;<br/>
<a NAME="E10:66_1"/><i><font color="Green">E55</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">dF</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="66_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="E1:66_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>then </b><i><font color="Green">E56</font></i>:  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">dF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> = 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="partfun1.html#K1">*</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="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon">d</font> <a href="relset_1.html#K10">.:</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="relat_1.html#T160">RELAT_1:160</a></i>
;<br/>
<a NAME="E3:66_1_1"/>
 <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/>
<b>hence </b><a NAME="E4:66_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">dF</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>, , <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">gdF</font> = <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">g</font></span>)</span> <a href="msualg_3.html#K3">**</a> <font color="Maroon">dF</font> as   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<font color="Maroon">S</font> ;<br/>
<i><font color="Green">E77</font></i>: <font color="Maroon">F</font> = 
<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> <span class="p2">(<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">S</font></span>)</span></span>)</span> <a href="pboole.html#K12">**</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="msualg_3.html#T4">MSUALG_3:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">g</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="pboole.html#K12">**</a> <font color="Maroon">F</font>
<b>by </b><i>, <a class="txt" href="waybel_5.html#E1:66_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">gdF</font>
<b>by </b><i><a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>
;<br/>
<a NAME="E13:66_1"/><i><font color="Green">E78</font></i>: 
 <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="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:66_1_3"/>
<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"><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="E3:66_1_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i/>;<br/>
<font color="Maroon">gdF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">g</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">dF</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="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">dF</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="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E5:66_1_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">gdF</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">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">dF</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="relat_1.html#T160">RELAT_1:160</a></i>;<br/>
<a NAME="E6:66_1_3"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">dF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E7:66_1_3"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">dF</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/>
<a NAME="E8:66_1_3"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">g</font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">dF</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#E3:66_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E9:66_1_3"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">dF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E10:66_1_3"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">dF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">dF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<a NAME="E11:66_1_3"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" target="_self">Sup</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E8:66_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E12:66_1_3"/>
(  <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp;  <a href="yellow_2.html#NK6" target="_self">Sup</a>  <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/>
<b>hence </b><a NAME="E13:66_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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="txt" href="waybel_5.html#E10:66_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:66_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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="tarski.html#R1">c=</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="66_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:66_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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>
 ;<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="E3:66_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:66_1_4"/><i><font color="Green">E98</font></i>: 
<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>and </b><br/><a NAME="E5:66_1_4"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>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="E7:66_1_4"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E8:66_1"><i><font color="Green">E53</font></i></a>, </i>;<br/>
<a NAME="E8:66_1_4"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">dF</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/>
<a NAME="E9:66_1_4"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">g</font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">dF</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#E3:66_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E10:66_1_4"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">dF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E11:66_1_4"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">dF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">dF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<a NAME="E12:66_1_4"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">dF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="waybel_5.html#E10:66_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<font color="Maroon">gdF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">g</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">dF</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="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">dF</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="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E14:66_1_4"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">gdF</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">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">dF</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="relat_1.html#T160">RELAT_1:160</a></i>;<br/>
<a NAME="E15:66_1_4"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E16:66_1_4"/>
<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"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E15:66_1"/><i><font color="Green">E103</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</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>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E16:66_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">g</font> <a href="waybel_0.html#R3">preserves_inf_of</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><a class="txt" href="waybel_5.html#E3:66_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>
<a NAME="E17:66_1"/>
(  <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> is <a href="waybel_0.html#V1">directed</a> &amp; not  <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> is <a href="xboole_0.html#V1">empty</a> )
 
<b>by </b><i>, <a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a></i>;<br/>
<a NAME="E18:66_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">g</font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <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>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:66_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E19:66_1"/><i><font color="Green">E110</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <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>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E20:66_1"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">f</font> being   <a href="hidden.html#M1">set</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">dF</font></span>)</span> holds <br/> <a href="relat_1.html#K2">rng</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> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:66_1_5"/><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">dF</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</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">dF</font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green">E114</font></i>:  <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">dF</font></span>)</span> = 
 <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">dF</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <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/>
<br/>.= 
 <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>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<i><font color="Green">E115</font></i>:  <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" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">dF</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<i><font color="Green">E116</font></i>:  <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> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
<b>by </b><i>, <a class="ref" href="waybel_5.html#T6" target="_self">Th6</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/>
<a NAME="E6:66_1_5"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:66_1_5"/><b>then </b>
 <a href="relset_1.html#K5">rng</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">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
;<br/>
<a NAME="E8:66_1_5"/><b>then </b><i><font color="Green">E117</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_5_4"><b>now </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="E1:66_1_5_4"/><i><font color="Green">E118</font></i>: 
<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">j</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> ;<br/><a NAME="E3:66_1_5_4"/><i><font color="Green">E119</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">dF</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:66_1_5_4"/><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">dF</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:66_1_5_4"/><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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="partfun1.html#K1">*</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="msualg_3.html#T2">MSUALG_3:2</a></i>;<br/><a NAME="E6:66_1_5_4"/><b>then </b><i><font color="Green">E120</font></i>: 
<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">d</font> <a href="partfun1.html#K1">*</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="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E7:66_1_5_4"/><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="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dF</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="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">J</font> <a href="waybel_5.html#K8" target="_self">=&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<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> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<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> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<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></span>)</span> <a href="funct_1.html#K1">.</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>
<b>by </b><i><a class="ref" href="waybel_5.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K1">*</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> <a href="funct_1.html#K1">.</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>
<b>by </b><i>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<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="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
<br/>.= 
<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>
<b>by </b><i>, <a class="ref" href="waybel_5.html#T6" target="_self">Th6</a>, <a class="ref" href="waybel_5.html#T9" target="_self">Th9</a>, </i>
;<br/><b>hence </b><a NAME="E9:66_1_5_4"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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" 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">i</font>
 ;<br/></div><b>end;</b></div>
<b>then </b> <a href="relat_1.html#K2">rng</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> = 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="waybel_5.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T160">RELAT_1:160</a></i>
;<br/>
<b>hence </b><a NAME="E11:66_1_5"/>
 <a href="relat_1.html#K2">rng</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> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E21:66_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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#NK9" target="_self">Infs</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:66_1_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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#NK9" target="_self">Infs</a> </span>)</span></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="E3:66_1_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:66_1_6"/><i><font color="Green">E123</font></i>: 
<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#NK9" target="_self">Infs</a> </span>)</span>
 <b>and </b><br/><a NAME="E5:66_1_6"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:66_1_6"/><i><font color="Green">E125</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">dF</font></span>)</span>
 <b>and </b><br/><a NAME="E8:66_1_6"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <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">dF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_5.html#T6" target="_self">Th6</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</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">dF</font></span>)</span> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</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/>;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="relset_1.html#K5">rng</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">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span>;<br/>
<a NAME="E11:66_1_6"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">g</font> <a href="waybel_0.html#R3">preserves_inf_of</a>  <a href="relset_1.html#K5">rng</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">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:66_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>
<a NAME="E12:66_1_6"/>
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="relset_1.html#K5">rng</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">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E13:66_1_6"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/>
<a NAME="E14:66_1_6"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_5.html#T7" target="_self">Th7</a>, <a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>
<i><font color="Green">E129</font></i>:  <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">dF</font></span>)</span> = 
 <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">dF</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <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/>
<br/>.= 
 <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>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E16:66_1_6"/>
 <a href="relat_1.html#K2">rng</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> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:66_1_6"/><b>then </b>
<font color="Maroon">y</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="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E18:66_1_6"/>
<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"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_5.html#T8" target="_self">Th8</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:66_1"/><i><font color="Green">E130</font></i>: 
 <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="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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#NK9" target="_self">Infs</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:66_1_7"/>
<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"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:66_1_7"/><i><font color="Green">E140</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>
 <b>and </b><br/><a NAME="E4:66_1_7"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <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">S</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</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="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</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">dF</font></span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E142</font></i>:  <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">dF</font></span>)</span> = 
 <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">dF</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <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/>
<br/>.= 
 <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>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E8:66_1_7"/><b>then </b>
 <a href="relset_1.html#K5">rng</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="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:66_1_7"/><b>then </b><i><font color="Green">E143</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"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="relset_1.html#K5">rng</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">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f'</font></span>)</span>;<br/>
<a NAME="E10:66_1_7"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">g</font> <a href="waybel_0.html#R3">preserves_inf_of</a>  <a href="relset_1.html#K5">rng</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">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:66_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>
<a NAME="E11:66_1_7"/>
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="relset_1.html#K5">rng</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">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f'</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<b>then </b><i><font color="Green">E145</font></i>: <font color="Maroon">y</font> = 
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">dF</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>
;<br/>
<a NAME="E13:66_1_7"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <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#NK9" target="_self">Infs</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_5.html#T7" target="_self">Th7</a>, </i>;<br/>
<b>hence </b><a NAME="E14:66_1_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</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#NK9" target="_self">Infs</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_5.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
 <a href="yellow_2.html#NK7" target="_self">Inf</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_2.html#D6">YELLOW_2:def 6</a></i>
<br/>.= 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E8:66_1"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" target="_self">Sup</a> </span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></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">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E10:66_1"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
 <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"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_5.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E24:66_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> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:66"/>
<font color="Maroon">S</font> is  <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i/>;<br/>


</div>
