<?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_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:57"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">l</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Olive">x</font> is <a href="waybel_6.html#V6" target="_self">co-prime</a> ) )
 ;<br/>

<b>thus </b><a NAME="E2:57"/>
<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">V8</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="E3:57"/><i><font color="Green">E47</font></i>: 
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="waybel_5.html#T16">WAYBEL_5:16</a></i>;<br/>
<b>set </b><font color="Maroon">l</font> =  <a href="yellow_2.html#NK7" target="_self">Inf</a> ;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" target="_self">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="lattice3.html#K7">~</a> </span>)</span></span>)</span>;<br/>
<a NAME="E4:57"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" target="_self">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="lattice3.html#K7">~</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K1">waybelow</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="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" target="_self">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="lattice3.html#K7">~</a> </span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E6:57"/><i><font color="Green">E50</font></i>: 
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:57"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:57"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> is <a href="waybel_6.html#V6" target="_self">co-prime</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:57_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_6.html#K4" target="_self">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice3.html#K7">~</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:57_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#K8">~</a>  is <a href="waybel_6.html#V5" target="_self">prime</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:57_1"/>
<font color="Maroon">x</font> is <a href="waybel_6.html#V6" target="_self">co-prime</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:57"/><i><font color="Green">E55</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span></span>)</span> <a href="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/>
<a NAME="E10:57"/>
<font color="Maroon">X</font> <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="57_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <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="E1:57_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<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> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> );<br/>
<a NAME="E2:57_2"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> holds <br/> 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="Olive">j</font>,<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:57_2_1"/>
<font color="Maroon">j1</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">j1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> ;<br/>
<a NAME="E3:57_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">p</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<b>by </b><i>, , <a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/>
<a NAME="E4:57_2_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:57_2_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</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><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:57_2_1"/>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="waybel_5.html#D1">WAYBEL_5:def 1</a></i>;<br/>
<a NAME="E7:57_2_1"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i>, <a class="txt" href="waybel_6.html#E3:57"><i><font color="Green">E47</font></i></a>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E8:57_2_1"/>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="hidden.html#R1">=</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_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E10:57_2_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:57"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_6.html#E4:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel_3.html#T18">WAYBEL_3:18</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>;<br/>
<a NAME="E12:57_2_1"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">A</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E13:57_2_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <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"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E14:57_2_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">p</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"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<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_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E15:57_2_1"/>
<font color="Maroon">p</font> is <a href="waybel_6.html#V6" target="_self">co-prime</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E17:57_2_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E6:57"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1"><b>suppose </b></a><a NAME="E1:57_2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:57_2_1_1_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">k1</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">a</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">k1</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">k1</font> = <font color="Maroon">k1</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="waybel_6.html#E7:57"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E5:57_2_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><span class="p3">(<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_2.html#K3">.</a> <font color="Maroon">k1</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>
 <b>by </b><i>, <a class="txt" href="waybel_6.html#E7:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>hence </b><a NAME="E6:57_2_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">j1</font>,<font color="Olive">k</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_2"><b>suppose </b></a><a NAME="E1:57_2_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_2"/><b>then </b>
<font color="Maroon">a</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_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:57_2_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><span class="p3">(<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_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>hence </b><a NAME="E4:57_2_1_1_2"/>
 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">j1</font>,<font color="Olive">k</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:57_2"/><i><font color="Green">E100</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 <b>and </b><br/><a NAME="E5:57_2"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Maroon">f1</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="57_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:57_2_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 ;<br/><a NAME="E2:57_2_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/><a NAME="E3:57_2_2"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="waybel_6.html#E1:57"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_6.html#E4:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_6.html#T28">FUNCT_6:28</a></i>;<br/><b>hence </b><a NAME="E4:57_2_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_6.html#D2">FUNCT_6:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:57_2"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:57_2_3"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span>
 ;<br/><a NAME="E2:57_2_3"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_6.html#D2">FUNCT_6:def 2</a></i>;<br/><b>hence </b><a NAME="E3:57_2_3"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="txt" href="waybel_6.html#E1:57"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_6.html#T28">FUNCT_6:28</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:57_2"/><b>then </b><i><font color="Green">E105</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E10:57_2"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</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> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E4:57"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E11:57_2"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</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> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</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="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:57_2_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">b</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> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<a NAME="E2:57_2_4"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E1:57"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="waybel_5.html#T6">WAYBEL_5:6</a></i>;<br/>
<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">b</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="txt" href="waybel_6.html#E1:57"><i><font color="Green">E46</font></i></a>, , <a class="txt" href="waybel_6.html#E7:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="waybel_6.html#E8:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:57_2_4"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</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">b</font>
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:57"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_6.html#E3:57"><i><font color="Green">E47</font></i></a>, , <a class="txt" href="waybel_6.html#E7:57"><i><font color="Green">E52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:57_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R2">in</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="txt" href="waybel_6.html#E6:57"><i><font color="Green">E50</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f1</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="txt" href="waybel_6.html#E6:57"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E15:57_2"/>
<font color="Maroon">p</font> <a href="lattice3.html#R1">is_&lt;=_than</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">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:57_2_5"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:57_2_5"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i><a class="txt" href="waybel_6.html#E8:57"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E5:57_2_5"/>
( <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">Frege</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="txt" href="waybel_6.html#E7:57"><i><font color="Green">E52</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:57_2_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</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">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>;<br/>
<a NAME="E7:57_2_5"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E3:57"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:57_2_5"/>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="waybel_6.html#E8:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:57_2"/><b>then </b>
<font color="Maroon">p</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"><span class="p3">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<a NAME="E17:57_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> =  <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> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="waybel_6.html#E6:57"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">D</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">J</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">D</font> ;<br/>
<b>reconsider </b><font color="Maroon">R</font> =  <a href="waybel_5.html#K2">Frege</a> <font color="Maroon">F</font> as   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">P</font>,<font color="Maroon">L</font> ;<br/>
<a NAME="E22:57_2"/>
 <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></i>;<br/>
<a NAME="E23:57_2"/><b>then </b>
 <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#NK9" target="_self">Infs</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E24:57_2"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E25:57_2"/>
<font color="Maroon">p</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="txt" href="waybel_6.html#E8:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:57"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <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/>
<b>hence </b><a NAME="E12:57"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i>, , <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div>
