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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:45"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>3</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>5</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">b<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Olive">b<sub>2</sub></font></span>)</span> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Olive">b<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>8</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Olive">b<sub>7</sub></font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">b<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>8</sub></font>,<font color="Olive">b<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> ) &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  ) ) </span>}</span>  holds <br/> <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>4</sub></font>
 ;<br/>

<a NAME="E2:45"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>3</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st ( for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">b<sub>1</sub></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">b<sub>3</sub></font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">b<sub>4</sub></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="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be   <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>] means ex <font color="Olive">b<sub>1</sub></font> being <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> ) &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  ) );<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E3:45_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:45_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:45_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:45_1_1"/><i><font color="Green">E35</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; ex <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> ) &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  ) )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="relat_1.html#V2">V7</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:45_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:45_1_1"/><i><font color="Green">E37</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> ) &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  )
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:45_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:45_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E10:45_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E7:45_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</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">c<sub>4</sub></font></span>)</span> <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> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) );<br/>
<a NAME="E11:45_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E3:45_1_1_1"/><i><font color="Green">E41</font></i>: 
( not  <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">c<sub>4</sub></font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a> &amp;  <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">c<sub>4</sub></font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:45_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E4:45_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <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">c<sub>4</sub></font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E6:45_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel_5.html#E9:45_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel_5.html#E35"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:45_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:45_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E3:45_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:45_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E10:45_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</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">c<sub>4</sub></font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E6:45_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel_5.html#E9:45_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_5.html#T20" target="_self">Th20</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<a NAME="E12:45_1_1_1"/>
 <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">c<sub>9</sub></font> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E7:45_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E13:45_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<a NAME="E14:45_1_1_1"/>
 <a href="yellow_2.html#NK6" target="_self">Sup</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_5.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E15:45_1_1_1"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E16:45_1_1_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>
<a NAME="E17:45_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E10:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel_5.html#E13:45_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E18:45_1_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>14</sub></font>] )
 <b>by </b><i><a class="txt" href="waybel_5.html#E10:45_1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E13:45_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E14:45_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E15:45_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>(<a class="txt" href="waybel_5.html#E11:45_1_1"><i><font color="Green">E40</font></i></a>);<br/></i>
<a NAME="E16:45_1_1"/><i><font color="Green">E43</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">c<sub>4</sub></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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>;<br/>
<a NAME="E17:45_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></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">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E13:45_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel_5.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E18:45_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1_2"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel_5.html#E16:45_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_5.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E3:45_1_1_2"/>
 <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">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_5.html#T20" target="_self">Th20</a></i>;<br/>
<b>then </b><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">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>4</sub></font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="waybel_5.html#T20" target="_self">Th20</a></i>
;<br/>
<b>hence </b><a NAME="E5:45_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></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">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E15:45_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></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> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel_5.html#E17:45_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E20:45_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</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">c<sub>4</sub></font></span>)</span></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:45_1_1_3"/><i><font color="Green">E45</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> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><a href="waybel_5.html#K6" target="_self">curry</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:45_1_1_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_5.html#T20" target="_self">Th20</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:45_1_1_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</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">c<sub>4</sub></font></span>)</span> <a href="waybel_5.html#K1" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_5.html#E15:45_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:45_1_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</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">c<sub>4</sub></font></span>)</span></span>)</span> <a href="waybel_5.html#K3" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:45_1_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel_5.html#E2:45_1_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_5.html#E4:45_1_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_5.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:45_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#T57">YELLOW_2:57</a></i>;<br/>
<a NAME="E22:45_1_1"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_5.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E23:45_1_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E24:45_1_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E25:45_1_1"/>
<font color="Maroon">c<sub>6</sub></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="txt" href="waybel_5.html#E3:45_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel_5.html#E21:45_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:45_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E5:45_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:45"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E6:45_1"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 
<b>by </b><i><a class="ref" href="waybel_5.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E7:45_1"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel_5.html#E5:45_1"><i><font color="Green">E34</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 NAME="E3:45"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel_5.html#E32"><i><font color="Green">Lemma26</font></i></a></i>;<br/>


</div>
