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<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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:24_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">S</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:24_1"/><i><font color="Green">E28</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>  st 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> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><b>assume </b><a NAME="E3:24_1"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</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="E5:24_1"/><i><font color="Green">E30</font></i>: 
( <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> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</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>1</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> <font color="Maroon">S</font> );<br/><a NAME="E7:24_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>]
 <b>by </b><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="E9:24_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x0</font>]
 <b>and </b><br/><a NAME="E10:24_1"/><i><font color="Green">E37</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_1"/>
not 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> <font color="Maroon">S</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">z</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> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:24_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</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="ref" href="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</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="E6:24_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><b>hence </b><a NAME="E7:24_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:24_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:24"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">S</font> being   <a href="hidden.html#M1">set</a>   st ( 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>  st 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> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> ) holds <br/>the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font>
 ;<br/>

<b>assume </b><a NAME="E3:24"/>
not <font color="Maroon">R</font> is <a href="wellfnd1.html#V1" target="_self">well_founded</a>
 ;<br/>

<a NAME="E4:24"/><b>then </b>
not the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="wellord1.html#R1">is_well_founded_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i/>;<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="E6:24"/><i><font color="Green">E71</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>  )
 <b>and </b><br/><a NAME="E7:24"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or  not 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">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#D3">WELLORD1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </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>assume </b><a NAME="E1:24_2"/><i><font color="Green">E73</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font>
 ;<br/><b>assume </b><a NAME="E2:24_2"/>
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> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E3:24_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:24_2"/><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">x</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:24_2"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">z</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> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>thus </b><a NAME="E7:24_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:24"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:24"/><b>then </b>
<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> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E11:24"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T38">XBOOLE_1:38</a></i>;<br/>


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