<?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="E1:45"/><i><font color="Green" title="E32">A1</font></i>: 
for <font color="Olive" title="b1">J</font>, <font color="Olive" title="b2">K</font> being 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/> for <font color="Olive" title="b3">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">K</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b4">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b5">a</font> where <font color="Olive" title="b5">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b6">f</font> being <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="Olive" title="b1">J</font> st <br/>( <font color="Olive" title="b6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Olive" title="b1">J</font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Olive" title="b2">K</font></span>)</span> &amp;  ex <font color="Olive" title="b7">G</font> being  <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Olive" title="b6">f</font>,<font color="Maroon" title="c1">L</font> st <br/>( ( for <font color="Olive" title="b8">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">J</font><br/> for <font color="Olive" title="b9">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">j</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b7">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b8">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b8">j</font>,<font color="Olive" title="b9">x</font> ) &amp; <font color="Olive" title="b5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  ) ) </span>}</span>  holds <br/> <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_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b4">X</font>
 ;<br/>

<a NAME="E2:45"/>
for <font color="Olive" title="b1">J</font>, <font color="Olive" title="b2">K</font> being 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/> for <font color="Olive" title="b3">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">K</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>  st ( for <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">J</font> holds   <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b4">j</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) holds <br/> <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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">J</font>, <font color="Maroon" title="c3">K</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="c4">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">K</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><a NAME="E1:45_1"/><i><font color="Green" title="E33">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"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span> <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/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>] <b>means </b> ex <font color="Olive" title="b1">f</font> being <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> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">J</font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c3">K</font></span>)</span> &amp;  ex <font color="Olive" title="b2">G</font> being  <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">L</font> st <br/>( ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font><br/> for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b3">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">j</font>,<font color="Olive" title="b4">x</font> ) &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  ) );<br/>
<b>reconsider </b><font color="Maroon" title="c5">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E3:45_1"/>
<font color="Maroon" title="c5">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:45_1_1"/>
<font color="Maroon" title="c6">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">a</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="E3:45_1_1"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c6">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <b>and </b><br/><a NAME="E4:45_1_1"/><i><font color="Green" title="E35">A4</font></i>: 
 ex <font color="Olive" title="b1">f</font> being <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> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">J</font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c3">K</font></span>)</span> &amp;  ex <font color="Olive" title="b2">G</font> being  <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">L</font> st <br/>( ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font><br/> for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Olive" title="b3">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">j</font>,<font color="Olive" title="b4">x</font> ) &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">f</font> being  <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><b> such that </b><br/><a NAME="E6:45_1_1"/><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">J</font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c3">K</font></span>)</span>
 <b>and </b><br/><a NAME="E7:45_1_1"/><i><font color="Green" title="E37">A6</font></i>: 
 ex <font color="Olive" title="b1">G</font> being  <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c1">L</font> st <br/>( ( 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><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">G</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> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">x</font> ) &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  )
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:45_1_1"><i><font color="Green" title="E35">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">G</font> being   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E9:45_1_1"/><i><font color="Green" title="E38">A7</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><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">G</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> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">x</font>
 <b>and </b><br/><a NAME="E10:45_1_1"/><i><font color="Green" title="E39">A8</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E7:45_1_1"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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> &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"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span> <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="c7">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font> ) );<br/>
<a NAME="E11:45_1_1"/><i><font color="Green" title="E40">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1_1"/>
<font color="Maroon" title="c10">x</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="c11">j</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> ;<br/>
<a NAME="E3:45_1_1_1"/><i><font color="Green" title="E41">A10</font></i>: 
( not  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:45_1"><i><font color="Green" title="E33">A2</font></i></a></i>;<br/>
<a NAME="E4:45_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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"><span class="p3">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E6:45_1_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="waybel_5.html#E9:45_1_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="waybel_5.html#E35"><i><font color="Green" title="E28">Lm12</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">y</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="E6:45_1_1_1"/><i><font color="Green" title="E42">A11</font></i>: 
<font color="Maroon" title="c12">y</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"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span>
 <b>and </b><br/><a NAME="E7:45_1_1_1"/><i><font color="Green" title="E43">A12</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="c9">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E3:45_1_1_1"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="waybel_0.html#T1" title="WAYBEL_0:th.1">WAYBEL_0:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:45_1_1_1"/><i><font color="Green" title="E44">A13</font></i>: 
<font color="Maroon" title="c13">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"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span>
 <b>and </b><br/><a NAME="E10:45_1_1_1"/><i><font color="Green" title="E45">A14</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E6:45_1_1_1"><i><font color="Green" title="E42">A11</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="c14">k</font> = <font color="Maroon" title="c13">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">K</font> <b>by </b><i><a class="txt" href="waybel_5.html#E9:45_1_1_1"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="waybel_5.html#T20" target="_self" title="WAYBEL_5:th.20">Th20</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">k</font>
;<br/>

<a NAME="E12:45_1_1_1"/>
 <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="c9">G</font> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c11">j</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">y</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E7:45_1_1_1"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>
<a NAME="E13:45_1_1_1"/><b>then </b><i><font color="Green" title="E46">A15</font></i>: 
 <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> <font color="Maroon" title="c12">y</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E14:45_1_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="E15:45_1_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="E16:45_1_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="E17:45_1_1_1"/>
( <font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c14">k</font>] )
 <b>by </b><i><a class="txt" href="waybel_5.html#E10:45_1_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="waybel_5.html#E10:45_1_1_1"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="waybel_5.html#E13:45_1_1_1"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E13:45_1_1"/><i><font color="Green" title="E41">A16</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font>
 <b>and </b><br/><a NAME="E14:45_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">K</font>
 <b>and </b><br/><a NAME="E15:45_1_1"/><i><font color="Green" title="E42">A17</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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#E11:45_1_1"><i><font color="Green" title="E40">A9</font></i></a>);<br/></i>
<a NAME="E16:45_1_1"/><i><font color="Green" title="E43">A18</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</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/>
<a NAME="E17:45_1_1"/><b>then </b><i><font color="Green" title="E44">A19</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E13:45_1_1"><i><font color="Green" title="E41">A16</font></i></a>, <a class="ref" href="waybel_5.html#T20" target="_self" title="WAYBEL_5:th.20">Th20</a></i>;<br/>
<a NAME="E18:45_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">j</font> = <font color="Maroon" title="c11">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="txt" href="waybel_5.html#E16:45_1_1"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="waybel_5.html#T20" target="_self" title="WAYBEL_5:th.20">Th20</a></i>;<br/>
<a NAME="E3:45_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font>
 
<b>by </b><i><a class="ref" href="waybel_5.html#T20" target="_self" title="WAYBEL_5:th.20">Th20</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span>
<b>by </b><i><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>
<b>by </b><i><a class="ref" href="waybel_5.html#T20" target="_self" title="WAYBEL_5:th.20">Th20</a></i>
;<br/>
<b>hence </b><a NAME="E5:45_1_1_2"/>
<font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E15:45_1_1"><i><font color="Green" title="E42">A17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c10">g</font> as    <a href="pralg_2.html#M1" title="PRALG_2: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> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel_5.html#E17:45_1_1"><i><font color="Green" title="E44">A19</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E20:45_1_1"/>
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="c7">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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="E1:45_1_1_3"/><i><font color="Green" title="E45">A20</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:45_1_1_3"/><i><font color="Green" title="E46">A21</font></i>: 
<font color="Maroon" title="c2">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_5.html#T20" target="_self" title="WAYBEL_5:th.20">Th20</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">b</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="E4:45_1_1_3"/><i><font color="Green" title="E47">A22</font></i>: 
( <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="waybel_5.html#K1" title="WAYBEL_5:func.1">.</a> <font color="Maroon" title="c12">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="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span> &amp; <font color="Maroon" title="c7">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">b</font> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E15:45_1_1"><i><font color="Green" title="E42">A17</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:45_1_1_3"/>
<font color="Maroon" title="c7">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" title="WAYBEL_5:func.2">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" title="WAYBEL_5:func.6">curry</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="waybel_5.html#K3" title="WAYBEL_5:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:45_1_1_3"><i><font color="Green" title="E45">A20</font></i></a>, <a class="txt" href="waybel_5.html#E2:45_1_1_3"><i><font color="Green" title="E46">A21</font></i></a>, <a class="txt" href="waybel_5.html#E4:45_1_1_3"><i><font color="Green" title="E47">A22</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="E21:45_1_1"/><b>then </b><i><font color="Green" title="E45">A23</font></i>: 
<font color="Maroon" title="c7">a</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/>
<a NAME="E22:45_1_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="E23:45_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_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="E24:45_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#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E25:45_1_1"/>
<font color="Maroon" title="c6">a'</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#E3:45_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="waybel_5.html#E21:45_1_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:45_1"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">X</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="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>
<a NAME="E5:45_1"/><b>then </b><i><font color="Green" title="E34">A24</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#E1:45"><i><font color="Green" title="E32">A1</font></i></a></i>;<br/>
<a NAME="E6:45_1"/>
 <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="E7:45_1"/>
 <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#E5:45_1"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel_5.html#E32"><i><font color="Green" title="E26">Lm10</font></i></a></i>;<br/>


</div>
