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

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon">c</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">r</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[] <b>means </b>for <font color="Olive">V</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">H</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Olive">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Olive">V</font>  ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Olive">V</font> st <font color="Olive">F</font> <a href="wellfnd1.html#R1" target="_self">is_recursively_expressed_by</a> <font color="Olive">H</font>;<br/>
<b>thus </b><a NAME="E1:27"/>
( <font color="Maroon">R</font> is <a href="wellfnd1.html#V1" target="_self">well_founded</a> implies <font color="Maroon">S<sub>1</sub></font>[] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">R</font> is <a href="wellfnd1.html#V1" target="_self">well_founded</a>
 ;<br/>

<b>let </b><font color="Maroon">V</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">H</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">V</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font>] <b>means </b>(  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">a<sub>1</sub></font> is <a href="waybel_0.html#V12">lower</a> &amp; ( 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="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Olive">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> ) );<br/>
<b>consider </b><font color="Maroon">fs</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E3:27_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">f</font> being  <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font> holds <br/> ( <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fs</font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">f</font>] )
 <b>from </b><i><a class="ref" href="wellfnd1.html#S1" target="_self">WELLFND1:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>assume </b><a NAME="E1:27_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fs</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fs</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font>, <font color="Maroon">gg</font> = <font color="Maroon">g</font> as   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font> <b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/><b>assume </b><a NAME="E3:27_1_1"/>
not <font color="Maroon">f</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">g</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:27_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">gg</font></span>)</span> &amp; <font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#D6">PARTFUN1:def 6</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">R</font> <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <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>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">gg</font> &amp; <font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E7:27_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>]
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>consider </b><font color="Maroon">x0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E9:27_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x0</font>]
 <b>and </b><br/><a NAME="E10:27_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  holds <br/>(  not <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> or  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">y</font>] or  not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">x0</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> )
 <b>from </b><i><a class="ref" href="wellfnd1.html#S2" target="_self">WELLFND1:sch 2</a>(, );<br/></i><a NAME="E11:27_1_1"/>
<font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">fr</font> = <font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">gr</font> = <font color="Maroon">gg</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:27_1_1_1"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span>
 ;<br/>

<a NAME="E2:27_1_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font> is <a href="waybel_0.html#V12">lower</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:27_1_1_1"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font> <a href="tarski.html#R1">c=</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:27_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="wellfnd1.html#K1" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E5:27_1_1_1"/><i><font color="Green">E72</font></i>: 
 <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">gg</font> is <a href="waybel_0.html#V12">lower</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:27_1_1_1"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font> <a href="tarski.html#R1">c=</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">gg</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:27_1_1_1"/><b>then </b>
 <a href="wellfnd1.html#K1" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">gg</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E1:27_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:27_1_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">gg</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E10:27_1_1_1"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x0</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E1:27_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="wellfnd1.html#E3:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i><a class="txt" href="wellfnd1.html#E3:27_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E12:27_1_1_1"/><i><font color="Green">E101</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">gg</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> )
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E1:27_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="wellfnd1.html#E3:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E13:27_1_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E1:27_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="wellfnd1.html#E3:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E14:27_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">gg</font> )
 
<b>by </b><i>, , <a class="ref" href="wellfnd1.html#T2" target="_self">Th2</a>, , </i>;<br/>
<b>hence </b><a NAME="E15:27_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="wellfnd1.html#E3:27_1"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="wellfnd1.html#E1:27_1_1"><i><font color="Green">E30</font></i></a>, </i>;<br/>


</div><b>end;</b></div><b>then </b><font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font> = 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">gg</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E13:27_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ufs</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">fs</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/>
<a NAME="E6:27_1"/><i><font color="Green">E104</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:27_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font>
 ;<br/>

<a NAME="E2:27_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ufs</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">fx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_1_2"/><i><font color="Green">E105</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> &amp; <font color="Maroon">fx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fs</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">ff</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:27_1_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fx</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<a NAME="E7:27_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:27_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:27_1"/><i><font color="Green">E107</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ufs</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:27_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ufs</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_3"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ufs</font> <a href="funct_1.html#K1">.</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/>
<a NAME="E4:27_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ufs</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">fx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:27_1_3"/><i><font color="Green">E109</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> &amp; <font color="Maroon">fx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fs</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">ff</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:27_1_3"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fx</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<a NAME="E9:27_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ff</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:27_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E111</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_4"><b>now </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="E1:27_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font>
 ;<br/><a NAME="E2:27_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ufs</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then consider </b><font color="Maroon">fx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_1_4"/><i><font color="Green">E112</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> &amp; <font color="Maroon">fx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fs</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">fx</font> as   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/><a NAME="E6:27_1_4"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E7:27_1_4"/>
 <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font> is <a href="waybel_0.html#V12">lower</a>
 <b>by </b><i>, </i>;<br/><a NAME="E8:27_1_4"/><b>then </b><i><font color="Green">E114</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font>
 <b>by </b><i>, </i>;<br/><a NAME="E9:27_1_4"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ufs</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>thus </b><font color="Maroon">ufs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a>, , </i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E116</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_5"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> )
 ;<br/><a NAME="E2:27_1_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ufs</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then consider </b><font color="Maroon">fx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_1_5"/><i><font color="Green">E118</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> &amp; <font color="Maroon">fx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fs</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">fx</font> as   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font> <b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a>, <a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/><a NAME="E6:27_1_5"/><i><font color="Green">E119</font></i>: 
 <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font> is <a href="waybel_0.html#V12">lower</a>
 <b>by </b><i>, <a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E7:27_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E8:27_1_5"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E9:27_1_5"/>
<font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ufs</font>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:27_1_5"/><b>then </b>
 <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">ff</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font>
 <b>by </b><i><a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><b>hence </b><a NAME="E11:27_1_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:27_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:27_1"/><i><font color="Green">E121</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_6"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_6"/><i><font color="Green">E122</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</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">R</font> <b>by </b><i>, <a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font> );<br/>
<a NAME="E5:27_1_6"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>]
 
<b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">x0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E7:27_1_6"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x0</font>]
 <b>and </b><br/><a NAME="E8:27_1_6"/><i><font color="Green">E125</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  holds <br/>(  not <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> or  not <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">y</font>] or  not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">x0</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> )
 <b>from </b><i><a class="ref" href="wellfnd1.html#S2" target="_self">WELLFND1:sch 2</a>(, );<br/></i>
<b>set </b><font color="Maroon">nv</font> = <font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p3">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p4">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon">nf</font> = <font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>;<br/>
<a NAME="E9:27_1_6"/><i><font color="Green">E128</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p5">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p5">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E10:27_1_6"/><i><font color="Green">E129</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x0</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E11:27_1_6"/><b>then </b>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:27_1_6"/><b>then </b>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p5">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E3:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E13:27_1_6"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p4">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p5">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p0"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E14:27_1_6"/><i><font color="Green">E131</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p5">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_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="E1:27_1_6_1"/><i><font color="Green">E132</font></i>: 
( <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"><font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p5">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span></span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 ;<br/>

<a NAME="E2:27_1_6_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:27_1_6_1"/><b>then </b>
<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"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E3:27_1"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:27_1_6_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:27_1_6"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p5">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_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="E1:27_1_6_2"/><i><font color="Green">E133</font></i>: 
<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"><font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p5">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:27_1_6_2"/><i><font color="Green">E134</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p5">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">ufs</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p4"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p5">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p0">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<b>assume </b><a NAME="E3:27_1_6_2"/><i><font color="Green">E135</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E4:27_1_6_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ufs</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:27_1_6_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<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"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:27_1_6_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E7:27_1_6_2"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x0</font>,<span class="p1">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:27_1_6_2"/>
<font color="Maroon">ufs</font> <a href="hidden.html#R2">in</a>  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<a NAME="E9:27_1_6_2"/><b>then </b>
<font color="Maroon">ufs</font> is   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/>
<a NAME="E10:27_1_6_2"/><b>then </b>
<font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span> is   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T44">PARTFUN1:44</a></i>;<br/>
<a NAME="E11:27_1_6_2"/><b>then </b>
<font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<b>hence </b><a NAME="E12:27_1_6_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="binop_1.html#T29">BINOP_1:29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:27_1_6"/><b>then </b>
<font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">nf</font> = <font color="Maroon">ufs</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span> as   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font> <b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/>
<a NAME="E18:27_1_6"/><i><font color="Green">E138</font></i>: 
 <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">nf</font> is <a href="waybel_0.html#V12">lower</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellfnd1.html#D1" target="_self">WELLFND1:def 1</a><br/>

<b>assume </b><a NAME="E1:27_1_6_3"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">nf</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> )
 ;<br/>

<b>assume </b><a NAME="E2:27_1_6_3"/><i><font color="Green">E140</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">nf</font>
 ;<br/>

<a NAME="E3:27_1_6_3"/><b>then </b><i><font color="Green">E141</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E3:27_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:27_1_6_3"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:27_1_6_3"/><b>then </b>
<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"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E3:27_1"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:27_1_6_3"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E8:27_1_6_3"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E3:27_1"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:27_1_6_3"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:27_1_6_3"/>
contradiction
 <b>by </b><i><a class="ref" href="wellfnd1.html#T2" target="_self">Th2</a>, , , </i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_4"><b>now </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="E1:27_1_6_4"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">nf</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_6_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_6_4_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font> or <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"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="wellfnd1.html#E3:27_1"><i><font color="Green">E28</font></i></a>, , <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="27_1_6_4_1_1"><b>suppose </b></a><a NAME="E1:27_1_6_4_1_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ufs</font>
 ;<br/><div class="add"><a NAME="E2:27_1_6_4_1_1"/><b>then </b><i><font color="Green">E145</font></i>: 
not <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"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:27_1"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:27_1_6_4_1_1"/><i><font color="Green">E146</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x0</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_4_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_6_4_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x0</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_6_4_1_1_1"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x0</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:27_1_6_4_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:27_1_6_4_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:27_1_6_4_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="wellfnd1.html#E1:27_1"><i><font color="Green">E26</font></i></a>, </i>;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">nf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">ufs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">nf</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T76">FUNCT_4:76</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_6_4_1_2"><b>suppose </b></a><a NAME="E1:27_1_6_4_1_2"/><i><font color="Green">E148</font></i>: 
<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"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:27_1_6_4_1_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:27_1_6_4_1_2"/><i><font color="Green">E150</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x0</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_6_4_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_6_4_1_2_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x0</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_1_6_4_1_2_1"/><i><font color="Green">E151</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"><font color="Maroon">x0</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:27_1_6_4_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:27_1_6_4_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon">nf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p4">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p5">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x0</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">nf</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , , <a class="ref" href="funct_4.html#T76">FUNCT_4:76</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E20:27_1_6"/><b>then </b>
<font color="Maroon">nf</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fs</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E21:27_1_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x0</font>,<span class="p2">(<span class="default"><font color="Maroon">nf</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ufs</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E22:27_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:27_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ufs</font> = <font color="Maroon">ufs</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">ufs</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellfnd1.html#D5" target="_self">WELLFND1:def 5</a><br/>

<b>thus </b><a NAME="E12:27_1"/>
<font color="Maroon">ufs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">ufs</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:27"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[]
 ;<br/>

<b>reconsider </b><font color="Maroon">ac</font> = <span class="p1">(<span class="default"><a href="card_1.html#K3">alef</a> 0</span>)</span> <a href="card_2.html#K1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span> as  <a href="finset_1.html#V1">infinite</a> <a href="card_1.html#NM1">Cardinal</a> ;<br/>
<b>set </b><font color="Maroon">V</font> =  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,   <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E4:27"/><i><font color="Green">E154</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><br/> for <font color="Olive">p</font> being   <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font></span>)</span> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">p</font>) <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font></span>)</span>;<br/>



<a NAME="E1:27_2"/>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">p</font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">p</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_1.html#T28">CARD_1:28</a></i>;<br/>
<a NAME="E2:27_2"/><b>then </b><i><font color="Green">E155</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:27_2"/>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">ac</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T72">CARD_4:72</a></i>;<br/>
<a NAME="E4:27_2"/><b>then </b><i><font color="Green">E156</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">ac</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:27_2"/>
<font color="Maroon">ac</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T32">CARD_1:32</a></i>;<br/>
<a NAME="E6:27_2"/><b>then </b><i><font color="Green">E157</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E7:27_2"/>
 <a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font> is <a href="card_5.html#V1">regular</a>
 
<b>by </b><i><a class="ref" href="card_5.html#T42">CARD_5:42</a></i>;<br/>
<b>hence </b><a NAME="E8:27_2"/>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">H</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<span class="p3">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font><b> such that </b><br/><a NAME="E6:27"/><i><font color="Green">E158</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><br/> for <font color="Olive">p</font> being   <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font></span>)</span> holds  <font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">p</font>)
 <b>from </b><i><a class="ref" href="funct_7.html#S1">FUNCT_7:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">rk</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font><b> such that </b><br/><a NAME="E8:27"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">rk</font> <a href="wellfnd1.html#R1" target="_self">is_recursively_expressed_by</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>
<a NAME="E9:27"/><i><font color="Green">E161</font></i>: 
 <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">rk</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a>,<a class="ref" href="wellfnd1.html#D2" target="_self">WELLFND1:def 2</a><br/>

<b>assume </b><a NAME="E10:27"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:27"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="ordinal2.html#K2">inf</a> <span class="p1">(<span class="default"><font color="Maroon">rk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span>;<br/>
<a NAME="E13:27"/>
<font color="Maroon">rk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font>
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E14:27"/><b>then </b>
( <font color="Maroon">rk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">rk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> is   <a href="ordinal1.html#NM2">Ordinal</a> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E15:27"/><b>then </b>
 <a href="ordinal2.html#K2">inf</a> <span class="p1">(<span class="default"><font color="Maroon">rk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">rk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T25">ORDINAL2:25</a></i>;<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="E17:27"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">rk</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">rk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K2">inf</a> <span class="p1">(<span class="default"><font color="Maroon">rk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E18:27"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a></i>;<br/>

<b>assume </b><a NAME="E19:27"/>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:27"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E22:27"/><i><font color="Green">E167</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E23:27"/><b>then </b>
<font color="Maroon">rk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">rke</font> = <font color="Maroon">rk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E25:27"/>
<font color="Maroon">rke</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E26:27"/><b>then </b><i><font color="Green">E169</font></i>: 
 <a href="ordinal2.html#K2">inf</a> <span class="p1">(<span class="default"><font color="Maroon">rk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">rk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T22">ORDINAL2:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">rkra</font> = <font color="Maroon">rk</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a</font></span>)</span> as    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,<span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <font color="Maroon">ac</font></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<i><font color="Green">E171</font></i>: <font color="Maroon">rk</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">rkra</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">rkra</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E30:27"/>
<font color="Maroon">rke</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">rkra</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T73">FUNCT_1:73</a></i>;<br/>
<a NAME="E31:27"/><b>then </b>
<font color="Maroon">rk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K2">inf</a> <span class="p1">(<span class="default"><font color="Maroon">rk</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a>, , <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E32:27"/>
contradiction
 <b>by </b><i><a class="ref" href="wellfnd1.html#T2" target="_self">Th2</a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>


</div>
