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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:40"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">J</font>, <font color="Olive">K</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">J</font>,<font color="Olive">K</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>  st ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">J</font> holds   <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Olive">F</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a> ) holds <br/> <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 ;<br/>

<a NAME="E2:40"/>
for <font color="Olive">J</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">K</font> being <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">J</font><br/> for <font color="Olive">F</font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Olive">K</font>,<font color="Maroon">L</font>  st ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">J</font> holds   <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a> ) holds <br/> <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">J</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">K</font> be  <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds   <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 ;<br/>

<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>;<br/>
<a NAME="E4:40_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 
;<br/>
<a NAME="E5:40_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:40_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">K</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">K'</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">K</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="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<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> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; ( ( <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; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<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> ) or ( not <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; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> ) ) );<br/>
<a NAME="E8:40_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">k'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K'</font>  ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Olive">k'</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/><b>let </b><font color="Maroon">k'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K'</font>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1_1"/>
( <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> or  not <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_1_1_1"/>
<font color="Maroon">k'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k'</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> ;<br/><b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">k</font>
;<br/><b>thus </b><a NAME="E3:40_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>,<font color="Maroon">k'</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">k</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_1_2"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>take </b>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
;<br/><b>thus </b><a NAME="E2:40_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>,<font color="Maroon">k'</font>, <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:40_1"/>
 ex <font color="Olive">G</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">K'</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <br/>for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K'</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Olive">k</font>,<font color="Olive">G</font> <a href="binop_1.html#K2">.</a> <font color="Olive">j</font>,<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<b>then consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">K'</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:40_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><br/> for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K'</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Olive">k</font>,<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Olive">j</font>,<font color="Olive">k</font>]
 ;<br/>
<a NAME="E12:40_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds  <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:40_1_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><a NAME="E3:40_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:40_1_2_1"/><b>then </b>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:40_1_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:40_1_2"/>
verum
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:40_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds   <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_3_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">k</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" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E4:40_1_3_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:40_1_3_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K'</font> )
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K'</font> ;<br/><a NAME="E7:40_1_3_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">K'</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E8:40_1_3_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:40_1_3_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> &amp;  <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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">K'</font></span>)</span> <a href="funct_1.html#K1">.</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="E10:40_1_3_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">k</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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><font color="Maroon">y</font> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="waybel_5.html#E1:40"><i><font color="Green">E34</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E1:40_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_5.html#T27">FUNCT_5:27</a></i>
;<br/><b>hence </b><a NAME="E12:40_1_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:40_1_3"/>
verum
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:40_1"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds   <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font>;<br/>

<b>set </b><font color="Maroon">X</font> =  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>;<br/>
<a NAME="E1:40_1_4"/>
for <font color="Olive">x</font>, <font color="Olive">y</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>  <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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Olive">y</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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> holds <br/> ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z</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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_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>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<a NAME="E1:40_1_4_1"/><i><font color="Green">E82</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:40_1_4_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</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="E4:40_1_4_1"/><i><font color="Green">E98</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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E5:40_1_4_1"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:40_1_4_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">b</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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E8:40_1_4_1"/><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</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>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K'</font> <b>by </b><i><a class="txt" href="waybel_5.html#E1:40_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel_5.html#E8:40_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E10:40_1_4_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">j</font>,<font color="Maroon">a'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">j</font>,<font color="Maroon">b'</font> )
 
<b>by </b><i><a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, , </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_4_1_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) or ( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) or ( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_4_1_1_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/><div class="add"><a NAME="E2:40_1_4_1_1_1"/><b>then </b><i><font color="Green">E104</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"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">b</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" target="_self">.</a> <font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:40_1_4_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> )
 <b>by </b><i>, <a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/><a NAME="E4:40_1_4_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:40_1_4_1_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E7:40_1_4_1_1_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:40"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">z</font>
;<br/><a NAME="E8:40_1_4_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1"><i><font color="Green">Lemma32</font></i></a></i>;<br/><b>hence </b><a NAME="E9:40_1_4_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E11:40_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_5.html#E12:40_1"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_4_1_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/><div class="add"><a NAME="E2:40_1_4_1_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">x</font>
;<br/><b>thus </b><a NAME="E3:40_1_4_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:40_1_4_1_1_3"/>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/><div class="add"><a NAME="E2:40_1_4_1_1_3"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">y</font>
;<br/><b>thus </b><a NAME="E3:40_1_4_1_1_3"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_4_1_1_4"><b>suppose </b></a><a NAME="E1:40_1_4_1_1_4"/>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/><div class="add"><a NAME="E2:40_1_4_1_1_4"/><b>then </b><i><font color="Green">E114</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> )
 <b>by </b><i>, <a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">x</font>
;<br/><b>thus </b><a NAME="E3:40_1_4_1_1_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_1_4"/>
 <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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:40_1"/><i><font color="Green">E115</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="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<a NAME="E16:40_1"/><i><font color="Green">E116</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="E17:40_1"/><i><font color="Green">E117</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<a NAME="E18:40_1"/><i><font color="Green">E118</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:40_1"/>
 <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> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1_5"/>
<font color="Maroon">x</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>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:40_1_5"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:40_1_5"/><i><font color="Green">E120</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span>
 <b>and </b><br/><a NAME="E5:40_1_5"/><i><font color="Green">E121</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>  <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">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E6:40_1_5"/><i><font color="Green">E122</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:40"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="waybel_5.html#E1:40_1"><i><font color="Green">E36</font></i></a>, , </i>;<br/>
<a NAME="E7:40_1_5"/>
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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_5_1"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:40_1_5_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:40_1"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</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_5.html#E1:40_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/>
<a NAME="E4:40_1_5_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:40"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="waybel_5.html#E11:40_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<i><font color="Green">E126</font></i>: <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <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" target="_self">.</a> <font color="Maroon">j'</font></span>)</span>
<b>by </b><i>, <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">j'</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E6:40_1_5_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K'</font>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j'</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
<font color="Maroon">K'</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E8:40_1_5_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E12:40_1"><i><font color="Green">E44</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:40_1_5"/>
<font color="Maroon">x</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> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E8:40_1"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:40_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E21:40_1"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E22:40_1"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> holds <br/> <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font></span>)</span>,<font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_6"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green">E140</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>
<br/>.= 
<font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<i><font color="Green">E141</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E8:40_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>
<br/>.= 
 <a href="funct_5.html#K1">proj1</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_5.html#D3">FUNCT_5:def 3</a></i>
<br/>.= 
 <a href="funct_5.html#K1">proj1</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">K'</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="funct_5.html#T11">FUNCT_5:11</a></i>
;<br/>
<a NAME="E5:40_1_6"/>
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/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_6_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_6_3"/>
<font color="Maroon">j'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> ;<br/>
<a NAME="E3:40_1_6_3"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K'</font> )
 
<b>by </b><i>, <a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">fj</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K'</font> ;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">fj</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>
<br/>.= 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">j</font>,<font color="Maroon">fj</font>
<b>by </b><i>, <a class="txt" href="waybel_5.html#E11:40_1"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="waybel_5.html#E8:40_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, </i>
;<br/>
<b>hence </b><a NAME="E6:40_1_6_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j'</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:40_1_6"/>
 <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f'</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f'</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:40_1"/><i><font color="Green">E143</font></i>: 
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_7_1"/>
( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds  <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K'</font> or  ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> st <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_7_1_1"><b>suppose </b></a><a NAME="E1:40_1_7_1_1"/><i><font color="Green">E144</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds  <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K'</font>
 ;<br/><div class="add"><a NAME="E2:40_1_7_1_1"/>
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/><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">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_7_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_7_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> ;<br/>
<i><font color="Green">E145</font></i>: <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <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" target="_self">.</a> <font color="Maroon">j'</font></span>)</span>
<b>by </b><i>, <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">j'</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j'</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
<font color="Maroon">K'</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:40_1_7_1_1_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:40_1_7_1_1"/><b>then </b>
 <a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K2">doms</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:40"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="waybel_5.html#E1:40_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E4:40_1_7_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E5:40_1_7_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E6:40_1_7_1_1"/>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_7_1_2"><b>suppose </b></a><a NAME="E1:40_1_7_1_2"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> st <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K'</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><b> such that </b><br/><a NAME="E3:40_1_7_1_2"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K'</font>
 ;<br/><a NAME="E4:40_1_7_1_2"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K'</font>
 <b>by </b><i/>;<br/><a NAME="E5:40_1_7_1_2"/><b>then </b>
not <font color="Maroon">K'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:40_1_7_1_2"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K'</font> &amp; not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K'</font> <b>by </b><i><a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E9:40_1_7_1_2"/><i><font color="Green">E148</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_7_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1_7_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_7_1_2_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_1_7_1_2_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="waybel_5.html#E12:40_1"><i><font color="Green">E44</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_7_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_7_1_2_1_1"/>
( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> or  ex <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> st not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_7_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_7_1_2_1_1_1"/><i><font color="Green">E151</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 ;<br/><div class="add"><i><font color="Green">E152</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E12:40_1"><i><font color="Green">E44</font></i></a>, </i>
<br/>.= 
 <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_5.html#E1:40"><i><font color="Green">E34</font></i></a>, , </i>
;<br/><a NAME="E3:40_1_7_1_2_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_7_1_2_1_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_7_1_2_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i>, <a class="ref" href="funct_6.html#T89">FUNCT_6:89</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">j</font> = 
 <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" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <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">j</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:40_1_7_1_2_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:40_1_7_1_2_1_1_1"/><b>then </b>
<font color="Maroon">f</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="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E5:40_1_7_1_2_1_1_1"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:40_1_7_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:40_1_7_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:40_1_7_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_7_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_7_1_2_1_1_2"/>
not for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font><b> such that </b><br/><a NAME="E3:40_1_7_1_2_1_1_2"/><i><font color="Green">E154</font></i>: 
not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E4:40_1_7_1_2_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><a NAME="E5:40_1_7_1_2_1_1_2"/><b>then </b>
<font color="Maroon">j</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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:40_1_7_1_2_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E12:40_1"><i><font color="Green">E44</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K'</font> <b>by </b><i/>;<br/> <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E9:40_1_7_1_2_1_1_2"/><b>then </b>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E12:40_1"><i><font color="Green">E44</font></i></a>, </i>;<br/><a NAME="E10:40_1_7_1_2_1_1_2"/><b>then </b>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p4">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/><a NAME="E11:40_1_7_1_2_1_1_2"/><b>then </b><i><font color="Green">E155</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/><a NAME="E12:40_1_7_1_2_1_1_2"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><a NAME="E13:40_1_7_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/><a NAME="E14:40_1_7_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</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"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E15:40_1_7_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E10:40_1_7_1_2"/><i><font color="Green">E156</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_7_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1_7_1_2_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_7_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_7_1_2_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> or <font color="Maroon">x</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"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</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="40_1_7_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:40_1_7_1_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_7_1_2_2_1_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span>
 <b>and </b><br/><a NAME="E4:40_1_7_1_2_2_1_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/><a NAME="E5:40_1_7_1_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:40_1_7_1_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E8:40_1"><i><font color="Green">E38</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_7_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:40_1_7_1_2_2_1_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">x</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"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:40_1_7_1_2_2_1_2"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">f</font> = <font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font>;<br/><i><font color="Green">E163</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font></span>)</span> = 
<font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E1:40_1"><i><font color="Green">E36</font></i></a>, </i>
;<br/><a NAME="E5:40_1_7_1_2_2_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_7_1_2_2_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_7_1_2_2_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">x</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_5.html#E1:40_1"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<a NAME="E3:40_1_7_1_2_2_1_2_2"/><i><font color="Green">E164</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E2"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
<font color="Maroon">K'</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:40_1_7_1_2_2_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E6:40_1_7_1_2_2_1_2"/><b>then </b>
<font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</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> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E7:40_1_7_1_2_2_1_2"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><font color="Maroon">x</font> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><a NAME="E9:40_1_7_1_2_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E10:40_1_7_1_2_2_1_2"/><b>then </b>
<font color="Maroon">x'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p4">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/><a NAME="E11:40_1_7_1_2_2_1_2"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">x'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/><a NAME="E12:40_1_7_1_2_2_1_2"/>
<font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><a NAME="E13:40_1_7_1_2_2_1_2"/><b>then </b>
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2" target="_self">Frege</a> <span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">k</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/><b>hence </b><a NAME="E14:40_1_7_1_2_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E11:40_1_7_1_2"/><i><font color="Green">E167</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</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="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E12:40_1_7_1_2"/><i><font color="Green">E168</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</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#T44">YELLOW_0:44</a></i>;<br/> <a href="yellow_2.html#NK6" target="_self">Sup</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#D5">YELLOW_2:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span> <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"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_5.html#E11:40_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_5.html#E12:40_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_2.html#T3">YELLOW_2:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_2.html#NK6" target="_self">Sup</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/><b>hence </b><a NAME="E14:40_1_7_1_2"/>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<i><font color="Green">E169</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> = 
<font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E25:40_1"/>
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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> holds <br/> <a href="yellow_2.html#K4">\\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K4">\\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span>,<font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_9"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_9"/>
<font color="Maroon">j'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j'</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <span class="p1">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">K'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E4:40_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">K'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K'</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E6:40_1_9"/><i><font color="Green">E171</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E7:40_1_9"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</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#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1"><i><font color="Green">Lemma32</font></i></a></i>;<br/>
<a NAME="E8:40_1_9"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E9:40_1_9"/><b>then </b>
 <a href="yellow_2.html#NK6" target="_self">Sup</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> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E10:40_1_9"/><b>then </b><i><font color="Green">E172</font></i>: 
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E11:40_1_9"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K'</font> holds  <font color="Maroon">H</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">k</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">K'</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_9_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> or  not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_9_1_1_1"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E174</font></i>: <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">H</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i/>
;<br/><a NAME="E3:40_1_9_1_1_1"/>
<font color="Maroon">k</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" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_5.html#E12:40_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:40_1_9_1_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:40_1_9_1_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/><b>hence </b><a NAME="E6:40_1_9_1_1_1"/>
<font color="Maroon">H</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">k</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_2.html#D5">YELLOW_2:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_9_1_1_2"/>
not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>then </b> <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> = 
<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">H</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">k</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E3:40_1_9_1_1_2"/>
<font color="Maroon">H</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">k</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#T44">YELLOW_0:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:40_1_9"/><b>then </b>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#T56">YELLOW_2:56</a></i>;<br/>
<b>hence </b><a NAME="E13:40_1_9"/>
 <a href="yellow_2.html#K4">\\/</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j'</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#K4">\\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j'</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E11:40_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b> <a href="yellow_2.html#NK7" target="_self">Inf</a>  = 
 <a href="yellow_2.html#NK7" target="_self">Inf</a> 
<b>by </b><i>, </i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i>, , <a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i/>;<br/>


</div>
