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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="wellfnd1.html#V1" target="_self">well_founded</a>  <a href="orders_2.html#L1">RelStr</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>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>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>let </b><font color="Maroon">F1</font> be   <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>, <font color="Maroon">F2</font> be   <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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">F1</font> <a href="wellfnd1.html#R1" target="_self">is_recursively_expressed_by</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">F2</font> <a href="wellfnd1.html#R1" target="_self">is_recursively_expressed_by</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E3:29"/><i><font color="Green">E30</font></i>: 
(  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp;  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">F2</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>assume </b><a NAME="E4:29"/>
<font color="Maroon">F1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E6:29"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</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> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">F1</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">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E8:29"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E9:29"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">R</font> is <a href="wellfnd1.html#V1" target="_self">well_founded</a>
 
;<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="E11:29"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x0</font>]
 <b>and </b><br/><a NAME="E12:29"/><i><font color="Green">E67</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>1</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="E13:29"/>
<font color="Maroon">F1</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">F2</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="29_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">fr</font> = <font color="Maroon">F1</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">F2</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="E1:29_1"/>
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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:29_1"/><b>then </b><i><font color="Green">E68</font></i>: 
(  <a href="wellfnd1.html#K1" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</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> &amp;  <a href="wellfnd1.html#K1" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</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/>
<b>assume </b><a NAME="E3:29_1"/>
not <font color="Maroon">F1</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">F2</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>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:29_1"/><i><font color="Green">E71</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">F1</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">F1</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">F2</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="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E6:29_1"/><i><font color="Green">E72</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#E1:29"><i><font color="Green">E26</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#E1:29"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E8:29_1"/><i><font color="Green">E73</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F1</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">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F2</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">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> )
 
<b>by </b><i>, <a class="txt" href="wellfnd1.html#E1:29"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E9:29_1"/>
<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#E1:29"><i><font color="Green">E26</font></i></a>, <a class="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:29_1"/>
contradiction
 <b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a>, <a class="txt" href="wellfnd1.html#E1:29"><i><font color="Green">E26</font></i></a>, <a class="ref" href="wellfnd1.html#T2" target="_self">Th2</a>, <a class="txt" href="wellfnd1.html#E2:29"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font> = 
<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">F2</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>, </i>
<br/>.= 
<font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x0</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E15:29"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
