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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>let </b><font color="Maroon">V</font> be  non <a href="realset1.html#V1">trivial</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>set </b><font color="Maroon">PF</font> =  <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>;<br/>
<b>set </b><font color="Maroon">wfp</font> =  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font>;<br/>
<b>consider </b><font color="Maroon">a0</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:28"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">a0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">a0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> )
 <b>by </b><i><a class="ref" href="realset1.html#T14">REALSET1:14</a></i>;<br/>
<b>set </b><font color="Maroon">a01</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a0</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K2">}</a></span>;<br/>
<a NAME="E3:28"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a0</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<b>set </b><font color="Maroon">F3</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font>;<br/>
<b>set </b><font color="Maroon">F4</font> = <span class="p1">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font>;<br/>
<b>set </b><font color="Maroon">F2</font> = <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span>;<br/>
<a NAME="E4:28"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span> <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="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E5:28"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<i><font color="Green">E35</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span></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"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
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#T12">XBOOLE_1:12</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">F1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</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="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E8:28"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a1</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a0</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E9:28"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span></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"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a0</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E10:28"/><b>then </b>
( <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span></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"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a0</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></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> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span></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"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a>, <a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E11:28"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a0</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:28"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F2</font> = <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span> 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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,  <a href="funct_1.html#NM1">Function</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>(  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> ) iff <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> );<br/>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_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>let </b><font color="Maroon">y</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>,<font color="Maroon">V</font>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">a1</font>, <font color="Maroon">v</font> = <font color="Maroon">a0</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a0</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K2">}</a></span> <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2_1"/>
(  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> ) or for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">y</font> or  not <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_1"><b>suppose </b></a><a NAME="E1:28_2_1_1"/><i><font color="Green">E39</font></i>: 
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">u</font> = <font color="Maroon">u</font>;<br/><b>thus </b><a NAME="E2:28_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">u</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1_2"><b>suppose </b></a><a NAME="E1:28_2_1_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">y</font> or  not <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">v</font> = <font color="Maroon">v</font>;<br/><b>thus </b><a NAME="E2:28_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">v</font>]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></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>,<font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a0</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E16:28"/><i><font color="Green">E68</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">y</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>,<font color="Maroon">V</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>();<br/></i>
<a NAME="E17:28"/><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">a0</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K2">}</a></span>
 
;<br/>
<a NAME="E18:28"/><i><font color="Green">E72</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <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> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as   <a href="funct_2.html#NM1">Function</a> of <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> <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/>
<a NAME="E20:28"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">F1</font> <a href="wellfnd1.html#R1" target="_self">is_recursively_expressed_by</a> <font color="Maroon">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellfnd1.html#D5" target="_self">WELLFND1:def 5</a><br/>

<b>reconsider </b><font color="Maroon">F1r</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">x</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>,<font color="Maroon">V</font> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_3_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">F1r</font>
 ;<br/><a NAME="E2:28_3_1"/><b>then </b>
( <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>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">F1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T86">RELAT_1:86</a></i>;<br/><b>then </b><font color="Maroon">F1r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">a0</font>
<b>by </b><i><a class="ref" href="wellfnd1.html#T2" target="_self">Th2</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E4:28_3_1"/>
<font color="Maroon">F1r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:28_3"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">F1r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:28_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">F1r</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:28_3"/><b>then </b><i><font color="Green">E102</font></i>: 
<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="Maroon">x</font>,<font color="Maroon">F1r</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">a0</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<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">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">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a>, <a class="ref" href="wellfnd1.html#T2" target="_self">Th2</a>, <a class="txt" href="wellfnd1.html#E3:28"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E21:28"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">F2</font> <a href="wellfnd1.html#R1" target="_self">is_recursively_expressed_by</a> <font color="Maroon">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4"><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellfnd1.html#D5" target="_self">WELLFND1:def 5</a><br/>

<b>reconsider </b><font color="Maroon">F2r</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">x</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>,<font color="Maroon">V</font> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_4_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_4_1_1"><b>suppose </b></a><a NAME="E1:28_4_1_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_4_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">F2r</font>
 ;<br/><a NAME="E2:28_4_1_1_1"/><b>then </b><i><font color="Green">E106</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>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">F2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T86">RELAT_1:86</a></i>;<br/><a NAME="E3:28_4_1_1_1"/><i><font color="Green">E107</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#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E3:28"><i><font color="Green">E28</font></i></a>, </i>;<br/><font color="Maroon">F2r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i>, , <a class="txt" href="wellfnd1.html#E4:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">a0</font>
<b>by </b><i>, <a class="txt" href="wellfnd1.html#E4:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E5:28_4_1_1_1"/>
<font color="Maroon">F2r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:28_4_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">H</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">F2r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font>
 <b>by </b><i/>;<br/><a NAME="E4:28_4_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">F2r</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><a NAME="E5:28_4_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<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="Maroon">x</font>,<font color="Maroon">F2r</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:28_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a0</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E3:28"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">a0</font>
<b>by </b><i>, <a class="txt" href="wellfnd1.html#E3:28"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<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">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">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="wellfnd1.html#T1" target="_self">Th1</a>, , <a class="txt" href="wellfnd1.html#E4:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_4_1_2"><b>suppose </b></a><a NAME="E1:28_4_1_2"/><i><font color="Green">E110</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:28_4_1_2"/><b>then </b>
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">x</font> <a href="tarski.html#R1">c=</a>  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</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="E4:28_4_1_2"/><i><font color="Green">E111</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; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:28_4_1_2"/><i><font color="Green">E112</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>
 <b>by </b><i/>;<br/><a NAME="E6:28_4_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1" target="_self">dom</a> <font color="Maroon">F2r</font>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T86">RELAT_1:86</a></i>;<br/><i><font color="Green">E114</font></i>: <font color="Maroon">F2r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">a1</font>
<b>by </b><i>, <a class="txt" href="wellfnd1.html#E4:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>thus </b><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="wellfnd1.html#E3:28"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<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">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">x</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>assume </b><a NAME="E22:28"/><i><font color="Green">E115</font></i>: 
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="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">V</font><br/> for <font color="Olive">F1</font>, <font color="Olive">F2</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="Maroon">V</font>  st <font color="Olive">F1</font> <a href="wellfnd1.html#R1" target="_self">is_recursively_expressed_by</a> <font color="Olive">H</font> &amp; <font color="Olive">F2</font> <a href="wellfnd1.html#R1" target="_self">is_recursively_expressed_by</a> <font color="Olive">H</font> holds <br/><font color="Olive">F1</font> <a href="funct_2.html#R2">=</a> <font color="Olive">F2</font>
 ;<br/>

<b>assume </b><a NAME="E23:28"/>
not <font color="Maroon">R</font> is <a href="wellfnd1.html#V1" target="_self">well_founded</a>
 ;<br/>

<a NAME="E24:28"/><b>then </b>
 <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</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>
 
<b>by </b><i/>;<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="E26:28"/><i><font color="Green">E116</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</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="wellfnd1.html#K2" target="_self">well_founded-Part</a> <font color="Maroon">R</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E27:28"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a0</font>
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a1</font></span>)</span> <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">a1</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E29:28"/>
contradiction
 <b>by </b><i>, <a class="txt" href="wellfnd1.html#E2:28"><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#E3:28"><i><font color="Green">E28</font></i></a>, <a class="txt" href="wellfnd1.html#E4:28"><i><font color="Green">E30</font></i></a></i>;<br/>


</div>
