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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">J</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">K</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V7</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">J</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><a NAME="E2:35"/><i><font color="Green" title="E24">A2</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> holds   <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E1:35_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 ;<br/><a NAME="E2:35_1"/><i><font color="Green" title="E26">A4</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> holds  <font color="Maroon" title="c5">c</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font>;<br/>

<a NAME="E1:35_1_1"/>
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_5.html#T14" target="_self" title="WAYBEL_5:th.14">Th14</a></i>;<br/>
<a NAME="E2:35_1_1"/><b>then </b>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24" title="YELLOW_2:th.24">YELLOW_2:24</a></i>;<br/>
<a NAME="E3:35_1_1"/><b>then </b>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#D6" title="YELLOW_2:def.6">YELLOW_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:35_1_1"/>
<font color="Maroon" title="c5">c</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:35_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:35_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font>  ex <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> &amp; <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">b</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:35_1_2"/><i><font color="Green" title="E27">A5</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font>  ex <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> st <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b2">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font>;<br/>

<a NAME="E1:35_1_2_1"/><i><font color="Green" title="E28">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E2:35"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<a NAME="E2:35_1_2_1"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 
<b>by </b><i><a class="txt" href="waybel_5.html#E2:35_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<a NAME="E3:35_1_2_1"/>
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:35_1_2_1"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span>
 <b>and </b><br/><a NAME="E6:35_1_2_1"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">d</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:35_1_2_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="waybel_5.html#E2:35_1_2_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="waybel_3.html#D1" title="WAYBEL_3:def.1">WAYBEL_3:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:35_1_2_1"/><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span>
 <b>and </b><br/><a NAME="E9:35_1_2_1"/><i><font color="Green" title="E33">A11</font></i>: 
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E5:35_1_2_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c8">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <b>by </b><i><a class="txt" href="waybel_5.html#E8:35_1_2_1"><i><font color="Green" title="E32">A10</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">k</font>
;<br/>

<b>thus </b><a NAME="E11:35_1_2_1"/>
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E6:35_1_2_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="waybel_5.html#E9:35_1_2_1"><i><font color="Green" title="E33">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp;  ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 &amp; <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font> ) );<br/>
<a NAME="E2:35_1_2"/><i><font color="Green" title="E28">A12</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">K</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">k</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:35_1_2_2"/>
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">j'</font> = <font color="Maroon" title="c6">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> ;<br/>
<b>consider </b><font color="Maroon" title="c8">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j'</font><b> such that </b><br/><a NAME="E4:35_1_2_2"/><i><font color="Green" title="E29">A13</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c7">j'</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:35_1_2"><i><font color="Green" title="E27">A5</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">k</font>
;<br/>

<a NAME="E5:35_1_2_2"/>
<font color="Maroon" title="c7">j'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font>
 
;<br/>
<a NAME="E6:35_1_2_2"/><b>then </b>
<font color="Maroon" title="c7">j'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E7:35_1_2_2"/><b>then </b>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j'</font> &amp; <font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">K</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:35_1_2_2"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>

<b>thus </b><a NAME="E9:35_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c8">k</font>]
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:35_1_2_2"><i><font color="Green" title="E29">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:35_1_2"/><i><font color="Green" title="E29">A14</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font>
 <b>and </b><br/><a NAME="E5:35_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <b>and </b><br/><a NAME="E6:35_1_2"/><i><font color="Green" title="E30">A15</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="waybel_5.html#E2:35_1_2"><i><font color="Green" title="E28">A12</font></i></a>);<br/></i>
<a NAME="E7:35_1_2"/><i><font color="Green" title="E31">A16</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> holds <br/> ( <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp;  ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> &amp; <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> ) )
 
<b>by </b><i><a class="txt" href="waybel_5.html#E6:35_1_2"><i><font color="Green" title="E30">A15</font></i></a></i>;<br/>
<i><font color="Green" title="E32">A17</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> = 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E4:35_1_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#T89" title="FUNCT_6:th.89">FUNCT_6:89</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_2_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/><a NAME="E2:35_1_2_4"/><b>then </b><i><font color="Green" title="E33">A18</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="funct_6.html#T89" title="FUNCT_6:th.89">FUNCT_6:89</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E2:35_1_2_4"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="funct_6.html#T31" title="FUNCT_6:th.31">FUNCT_6:31</a></i>
<br/>.= 
<font color="Maroon" title="c3">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/><b>hence </b><a NAME="E5:35_1_2_4"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E6:35_1_2"><i><font color="Green" title="E30">A15</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:35_1_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E8:35_1_2"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E11:35_1_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:35_1_2"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font>  ex <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> &amp; <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">b</font> ) ) )
 <b>by </b><i><a class="txt" href="waybel_5.html#E7:35_1_2"><i><font color="Green" title="E31">A16</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:35_1"/><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span>
 <b>and </b><br/><a NAME="E6:35_1"/><i><font color="Green" title="E28">A20</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font>  ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> &amp; <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <b>by </b><i><a class="txt" href="waybel_5.html#E5:35_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="waybel_5.html#E23"><i><font color="Green" title="E18">Lm7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">G</font> = <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c7">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">J</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font>;<br/><a NAME="E1:35_1_3"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font>
 ;<br/><a NAME="E2:35_1_3"/><b>then </b><i><font color="Green" title="E29">A21</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:35_1_3"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> &amp; <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E6:35_1"><i><font color="Green" title="E28">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E4:35_1_3"/>
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">G</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E5:35_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="waybel_5.html#E2:35_1_3"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="waybel_5.html#T9" target="_self" title="WAYBEL_5:th.9">Th9</a></i>;<br/></div><b>end;</b></div><a NAME="E10:35_1"/><b>then </b><i><font color="Green" title="E29">A22</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 <b>by </b><i><a class="ref" href="yellow_2.html#T57" title="YELLOW_2:th.57">YELLOW_2:57</a></i>;<br/><b>set </b><font color="Maroon" title="c9">a</font> =  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> ;<br/><a NAME="E11:35_1"/>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel_5.html#T14" target="_self" title="WAYBEL_5:th.14">Th14</a></i>;<br/><a NAME="E12:35_1"/><b>then </b>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_2.html#T24" title="YELLOW_2:th.24">YELLOW_2:24</a></i>;<br/><a NAME="E13:35_1"/><b>then </b>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <b>by </b><i><a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>;<br/><b>hence </b><a NAME="E14:35_1"/>
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E10:35_1"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:35"/><b>then </b><i><font color="Green" title="E25">A23</font></i>: 
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:35"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="waybel_5.html#T17" target="_self" title="WAYBEL_5:th.17">Th17</a></i>;<br/>
<a NAME="E5:35"/>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 
<b>by </b><i><a class="ref" href="waybel_5.html#T16" target="_self" title="WAYBEL_5:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E6:35"/>
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:35"><i><font color="Green" title="E25">A23</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/>


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