<?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>thus </b><a NAME="E1:61"/>
( <font color="Maroon">L</font> is <a href="waybel_5.html#V1">completely-distributive</a> implies 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">V3</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">L</font> holds   <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:61_1"/>
<font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a>
 <b>and </b><br/><a NAME="E2:61_1"/><i><font color="Green">E32</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> <br/> for <font color="Olive">K</font> being <a href="relat_1.html#V2">V3</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">L</font> holds   <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> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_5.html#D3">WAYBEL_5:def 3</a><br/>

<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">V3</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">L</font>;<br/>





<a NAME="E3:61_1"/><i><font color="Green">E33</font></i>: 
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<b>by </b><i><a class="txt" href="yellow_7.html#E3:61_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E4:61_1"/><i><font color="Green">E41</font></i>: 
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_7.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:61_1"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> &amp;  <a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="funct_6.html#K2">doms</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">J</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow_7.html#E2:61_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:61_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> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:61_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</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>reconsider </b><font color="Maroon">J'</font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">K'</font> = <font color="Maroon">J'</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J'</font> ;<br/>
<b>reconsider </b><font color="Maroon">G</font> =  <a href="waybel_5.html#K2">Frege</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> as   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K'</font>,<font color="Maroon">L</font> ;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J'</font><b> such that </b><br/><a NAME="E6:61_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</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="waybel_5.html#T14">WAYBEL_5:14</a></i>;<br/>
<a NAME="E7:61_1_1"/>
 ex <font color="Olive">g'</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">g'</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">g'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> holds <br/><font color="Olive">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 
<b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font></span>)</span></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<a NAME="E9:61_1_1"/><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">J'</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>  holds <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> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/><div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_1"><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="E1:61_1_1_1_1"/>
<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">i</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> ;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>);<br/><a NAME="E4:61_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow_7.html#E5:61_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E5:61_1_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>)
 ;<br/><b>hence </b><a NAME="E6:61_1_1_1_1"/>
 ex <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>,<font color="Olive">k</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:61_1_1_1"/><i><font color="Green">E52</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</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="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="txt" href="yellow_7.html#E1:61_1_1"><i><font color="Green">E43</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1_2"><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="E1:61_1_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><a NAME="E2:61_1_1_1_2"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>)
 <b>by </b><i><a class="txt" href="yellow_7.html#E6:61_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_1_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</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="yellow_7.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</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="yellow_7.html#T8" target="_self">Th8</a>, <a class="txt" href="yellow_7.html#E6:61_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E7:61_1_1_1"/>
<font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, <a class="txt" href="yellow_7.html#E4:61_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E9:61_1_1_1"/>
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>) &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>) )
 <b>by </b><i><a class="txt" href="yellow_7.html#E6:61_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><b>hence </b><a NAME="E10:61_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<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="E12:61_1_1"/><i><font color="Green">E111</font></i>: 
<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> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>) <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>
<a NAME="E13:61_1_1"/><i><font color="Green">E112</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"><font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</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">.</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="E14:61_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">.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:61_1_1_2"/>
<font color="Maroon">z</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">.</a> <font color="Maroon">j</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:61_1_1_2"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</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">u</font> = <font color="Maroon">u</font> as    <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><a class="txt" href="yellow_7.html#E6:61_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E5:61_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>)
 
<b>by </b><i><a class="txt" href="yellow_7.html#E5:61_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E7:61_1_1_2"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 ;<br/>
<a NAME="E8:61_1_1_2"/><i><font color="Green">E116</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> &amp; <font color="Maroon">K'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:61_1_1_2"/>
( <font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="pralg_1.html#K15">..</a> <font color="Maroon">g'</font> &amp; <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_1.html#K15">..</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D8">PRALG_2:def 8</a></i>;<br/>
<a NAME="E10:61_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_1.html#K15">..</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="yellow_7.html#T8" target="_self">Th8</a>, <a class="txt" href="yellow_7.html#E9:61_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/>
<a NAME="E11:61_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, <a class="txt" href="yellow_7.html#E6:61_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="yellow_7.html#E9:61_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E12:61_1_1_2"/>
<font color="Maroon">z</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">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</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="E15:61_1_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"><font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <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"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_7.html#E1:61_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>
<a NAME="E16:61_1_1"/><b>then </b>
( <font color="Maroon">a</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"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> &amp;  <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#NK7" target="_self">Infs</a> </span>)</span> )
 
<b>by </b><i>, <a class="ref" href="waybel_5.html#T14">WAYBEL_5:14</a>, <a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>
<a NAME="E17:61_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  &amp;  <a href="yellow_2.html#NK7" target="_self">Inf</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="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK7" target="_self">Infs</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a>, <a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>
<a NAME="E18:61_1_1"/><b>then </b>
<font color="Maroon">a</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="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK7" target="_self">Infs</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E19:61_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</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/>


</div><b>end;</b></div>
<a NAME="E7:61_1"/><b>then </b>
 <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> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E8:61_1"/><b>then </b>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:61_1"/>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <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_7.html#T7" target="_self">Th7</a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


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

<b>assume </b><a NAME="E2:61"/><i><font color="Green">E117</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> <br/> for <font color="Olive">K</font> being <a href="relat_1.html#V2">V3</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">L</font> holds   <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 ;<br/>

<b>thus </b><a NAME="E3:61"/>
<font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_5.html#D3">WAYBEL_5:def 3</a><br/>

<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">V3</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">L</font>;<br/>





<a NAME="E4:61"/><i><font color="Green">E118</font></i>: 
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="waybel_5.html#T16">WAYBEL_5:16</a></i>;<br/>
<a NAME="E5:61"/><i><font color="Green">E119</font></i>: 
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <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_7.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E6:61"/><i><font color="Green">E120</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> &amp;  <a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="funct_6.html#K2">doms</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">J</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:61"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK8" target="_self">Sups</a> </span>)</span> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">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="E1:61_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">a</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#NK8" target="_self">Sups</a> </span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">J'</font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">K'</font> = <font color="Maroon">J'</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J'</font> ;<br/>
<b>reconsider </b><font color="Maroon">G</font> =  <a href="waybel_5.html#K2">Frege</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> as   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K'</font>,<font color="Maroon">L</font> ;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J'</font><b> such that </b><br/><a NAME="E6:61_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">a</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="waybel_5.html#T14">WAYBEL_5:14</a></i>;<br/>
<a NAME="E7:61_2"/>
 ex <font color="Olive">g'</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">g'</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">g'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> holds <br/><font color="Olive">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 
<b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font></span>)</span></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> : <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<a NAME="E9:61_2"/><i><font color="Green">E123</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">J'</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">J</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_1"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>  holds <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> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/><div><i><font color="Green">E125</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_1"><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="E1:61_2_1_1"/>
<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">i</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> ;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>);<br/><a NAME="E4:61_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E5:61_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">i</font>)
 ;<br/><b>hence </b><a NAME="E6:61_2_1_1"/>
 ex <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>,<font color="Olive">k</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:61_2_1"/><i><font color="Green">E126</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</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="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_1_2"><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="E1:61_2_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><a NAME="E2:61_2_1_2"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>)
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:61_2_1_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</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="yellow_7.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</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="yellow_7.html#T8" target="_self">Th8</a>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E7:61_2_1"/>
<font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E9:61_2_1"/>
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>) &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</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> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:61_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<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="E12:61_2"/><i><font color="Green">E127</font></i>: 
<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> <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>) <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>
<a NAME="E13:61_2"/><i><font color="Green">E128</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"><font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">L</font> &amp;  <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">F</font> <a href="waybel_5.html#K1">.</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="E14:61_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:61_2_2"/>
<font color="Maroon">z</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">.</a> <font color="Maroon">j</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:61_2_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</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">u</font> = <font color="Maroon">u</font> as    <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/>
<a NAME="E5:61_2_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>)
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E7:61_2_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">g'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 ;<br/>
<a NAME="E8:61_2_2"/><i><font color="Green">E131</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> &amp; <font color="Maroon">K'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:61_2_2"/>
( <font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="pralg_1.html#K15">..</a> <font color="Maroon">g'</font> &amp; <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="pralg_1.html#K15">..</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D8">PRALG_2:def 8</a></i>;<br/>
<a NAME="E10:61_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pralg_1.html#K15">..</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="yellow_7.html#T8" target="_self">Th8</a>, , <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/>
<a NAME="E11:61_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, , , <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/>
<b>hence </b><a NAME="E12:61_2_2"/>
<font color="Maroon">z</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">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</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="E15:61_2"/><b>then </b>
 <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">G</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">g</font></span>)</span></span>)</span> <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="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E16:61_2"/><b>then </b>
( <font color="Maroon">a</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="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> &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>, <a class="ref" href="waybel_5.html#T14">WAYBEL_5:14</a>, <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E17:61_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a>  &amp;  <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</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#T24">YELLOW_2:24</a>, <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E18:61_2"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</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="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E19:61_2"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:61"/><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#NK8" target="_self">Sups</a> </span>)</span></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<a NAME="E9:61"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#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="E10:61"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i>, <a class="ref" href="yellow_7.html#T7" target="_self">Th7</a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div>
