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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">c</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">r</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c5">PF</font> =  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font>;<br/>
<b>set </b><font color="Maroon" title="c6">wfp</font> =  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font>;<br/>
<b>consider </b><font color="Maroon" title="c7">a0</font>, <font color="Maroon" title="c8">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:27"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c7">a0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c7">a0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a1</font> )
 <b>by </b><i><a class="ref" href="realset1.html#T14" title="REALSET1:th.14">REALSET1:14</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">a01</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">a0</font>,<font color="Maroon" title="c8">a1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<a NAME="E3:27"/><i><font color="Green" title="E18">A2</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">a0</font>,<font color="Maroon" title="c8">a1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E2:27"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">F3</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font>;<br/>
<b>set </b><font color="Maroon" title="c11">F4</font> = <span class="p1">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font>;<br/>
<b>set </b><font color="Maroon" title="c12">F2</font> = <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span>;<br/>
<a NAME="E4:27"/><i><font color="Green" title="E19">A3</font></i>: 
 <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E5:27"/><i><font color="Green" title="E20">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<i><font color="Green" title="E21">A5</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E4:27"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="wellfnd1.html#E5:27"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c13">F1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="wellfnd1.html#E2:27"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E8:27"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">a0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">a1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E9:27"/><b>then </b>
( <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">a0</font>,<font color="Maroon" title="c8">a1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a>, <a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></i>;<br/>
<a NAME="E10:27"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">a0</font>,<font color="Maroon" title="c8">a1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:27"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E3:27"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">F2</font> = <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="wellfnd1.html#E6:27"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>(  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> $2 &amp; $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font> ) iff $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font> );<br/>
<div><i><font color="Green" title="E22">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c16">y</font> be    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font>;<br/><b>reconsider </b><font color="Maroon" title="c17">u</font> = <font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c18">v</font> = <font color="Maroon" title="c7">a0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">a0</font>,<font color="Maroon" title="c8">a1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_1"/>
(  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c16">y</font> &amp; <font color="Maroon" title="c16">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font> ) or for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c16">y</font> or  not <font color="Maroon" title="c16">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1"><b>suppose </b></a><a NAME="E1:27_2_1_1"/><i><font color="Green" title="E23">A7</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c16">y</font> &amp; <font color="Maroon" title="c16">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c19">u</font> = <font color="Maroon" title="c17">u</font>;<br/><b>thus </b><a NAME="E2:27_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c19">u</font>]
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:27_2_1_1"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_2"><b>suppose </b></a><a NAME="E1:27_2_1_2"/><i><font color="Green" title="E23">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c16">y</font> or  not <font color="Maroon" title="c16">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c19">v</font> = <font color="Maroon" title="c18">v</font>;<br/><b>thus </b><a NAME="E2:27_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c19">v</font>]
 <b>by </b><i><a class="txt" href="wellfnd1.html#E2:27"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="wellfnd1.html#E1:27_2_1_2"><i><font color="Green" title="E23">A8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">a0</font>,<font color="Maroon" title="c8">a1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span><b> such that </b><br/><a NAME="E15:27"/><i><font color="Green" title="E23">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">y</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c15">H</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="wellfnd1.html#E13:27"><i><font color="Green" title="E22">A6</font></i></a>);<br/></i>
<a NAME="E16:27"/><i><font color="Green" title="E24">A10</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">a0</font>,<font color="Maroon" title="c8">a1</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
;<br/>
<a NAME="E17:27"/><i><font color="Green" title="E25">A11</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c15">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> )
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E3:27"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">H</font> = <font color="Maroon" title="c15">H</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E19:27"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c13">F1</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Maroon" title="c16">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellfnd1.html#D5" target="_self" title="WELLFND1:def.5">WELLFND1:def 5</a><br/>

<b>reconsider </b><font color="Maroon" title="c18">F1r</font> = <font color="Maroon" title="c13">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c17">x</font></span>)</span> as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:27_3_1"/><i><font color="Green" title="E27">A13</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c18">F1r</font>
 ;<br/><a NAME="E2:27_3_1"/><b>then </b>
( <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c17">x</font> &amp; <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c13">F1</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/><b>then </b><font color="Maroon" title="c18">F1r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> = 
<font color="Maroon" title="c13">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c7">a0</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E1:27_3_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E4:27_3_1"/>
<font color="Maroon" title="c18">F1r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a1</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E2:27"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:27_3"/><b>then </b><i><font color="Green" title="E27">A14</font></i>: 
<font color="Maroon" title="c16">H</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">F1r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a1</font>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E15:27"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>
<a NAME="E4:27_3"/><i><font color="Green" title="E28">A15</font></i>: 
<font color="Maroon" title="c16">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">F1r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">H</font>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E17:27"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c13">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font> = 
<font color="Maroon" title="c7">a0</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c16">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c17">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wellfnd1.html#E16:27"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="wellfnd1.html#E3:27_3"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="wellfnd1.html#E4:27_3"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E20:27"/><i><font color="Green" title="E27">A16</font></i>: 
<font color="Maroon" title="c14">F2</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Maroon" title="c16">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellfnd1.html#D5" target="_self" title="WELLFND1:def.5">WELLFND1:def 5</a><br/>

<b>reconsider </b><font color="Maroon" title="c18">F2r</font> = <font color="Maroon" title="c14">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c17">x</font></span>)</span> as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font> <b>by </b><i><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_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_4_1"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font> or  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_1_1"><b>suppose </b></a><a NAME="E1:27_4_1_1"/><i><font color="Green" title="E28">A17</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:27_4_1_1_1"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c18">F2r</font>
 ;<br/><a NAME="E2:27_4_1_1_1"/><b>then </b><i><font color="Green" title="E29">A18</font></i>: 
( <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c17">x</font> &amp; <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c14">F2</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/><a NAME="E3:27_4_1_1_1"/><i><font color="Green" title="E30">A19</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c17">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:27_4_1_1"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="wellfnd1.html#T5" target="_self" title="WELLFND1:th.5">Th5</a></i>;<br/><font color="Maroon" title="c18">F2r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> = 
<font color="Maroon" title="c14">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E2:27_4_1_1_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E5:27"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="wellfnd1.html#E2:27_4_1_1_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="txt" href="wellfnd1.html#E3:27_4_1_1_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c7">a0</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E2:27_4_1_1_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="txt" href="wellfnd1.html#E3:27_4_1_1_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E5:27_4_1_1_1"/>
<font color="Maroon" title="c18">F2r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a1</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E2:27"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:27_4_1_1"/><b>then </b><i><font color="Green" title="E29">A20</font></i>: 
<font color="Maroon" title="c16">H</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">F2r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a1</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E15:27"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/><a NAME="E4:27_4_1_1"/><i><font color="Green" title="E30">A21</font></i>: 
<font color="Maroon" title="c16">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c18">F2r</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">H</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E17:27"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:27_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c7">a0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a0</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E1:27_4_1_1"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><font color="Maroon" title="c14">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font> = 
<font color="Maroon" title="c7">a0</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E5:27"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="wellfnd1.html#E1:27_4_1_1"><i><font color="Green" title="E28">A17</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c16">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c17">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wellfnd1.html#E16:27"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="wellfnd1.html#E3:27_4_1_1"><i><font color="Green" title="E29">A20</font></i></a>, <a class="txt" href="wellfnd1.html#E4:27_4_1_1"><i><font color="Green" title="E30">A21</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_1_2"><b>suppose </b></a><a NAME="E1:27_4_1_2"/><i><font color="Green" title="E28">A22</font></i>: 
not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><a NAME="E2:27_4_1_2"/><b>then </b>
not the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c17">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T10" target="_self" title="WELLFND1:th.10">Th10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:27_4_1_2"/><i><font color="Green" title="E29">A23</font></i>: 
( <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c17">x</font> &amp; not <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E5:27_4_1_2"/><i><font color="Green" title="E30">A24</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c17">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T4" target="_self" title="WELLFND1:th.4">Th4</a></i>;<br/><a NAME="E6:27_4_1_2"/><b>then </b><i><font color="Green" title="E31">A25</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c18">F2r</font>
 <b>by </b><i><a class="txt" href="wellfnd1.html#E6:27"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="wellfnd1.html#E4:27_4_1_2"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/><i><font color="Green" title="E32">A26</font></i>: <font color="Maroon" title="c18">F2r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> = 
<font color="Maroon" title="c14">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E4:27_4_1_2"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E5:27"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="wellfnd1.html#E4:27_4_1_2"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c8">a1</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E4:27_4_1_2"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="wellfnd1.html#E5:27_4_1_2"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>thus </b><font color="Maroon" title="c14">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font> = 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E5:27"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="wellfnd1.html#E1:27_4_1_2"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c8">a1</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c16">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c17">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">F2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c17">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wellfnd1.html#E15:27"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="wellfnd1.html#E6:27_4_1_2"><i><font color="Green" title="E31">A25</font></i></a>, <a class="txt" href="wellfnd1.html#E7:27_4_1_2"><i><font color="Green" title="E32">A26</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>assume </b><a NAME="E21:27"/><i><font color="Green" title="E28">A27</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">V</font><br/> for <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">V</font>  st <font color="Olive" title="b2">F1</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b3">F2</font> <a href="wellfnd1.html#R1" title="WELLFND1:pred.1">is_recursively_expressed_by</a> <font color="Olive" title="b1">H</font> holds <br/><font color="Olive" title="b2">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">F2</font>
 ;<br/>

<b>assume </b><a NAME="E22:27"/>
not <font color="Maroon" title="c1">R</font> is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 ;<br/>

<a NAME="E23:27"/><b>then </b>
 <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#T9" target="_self" title="WELLFND1:th.9">Th9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E25:27"/><i><font color="Green" title="E29">A28</font></i>: 
( ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font> &amp; not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ) or ( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K2" title="WELLFND1:func.2">well_founded-Part</a> <font color="Maroon" title="c1">R</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E26:27"/><i><font color="Green" title="E30">A29</font></i>: 
<font color="Maroon" title="c13">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a0</font>
 
<b>by </b><i><a class="txt" href="wellfnd1.html#E25:27"><i><font color="Green" title="E29">A28</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<font color="Maroon" title="c14">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> = 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">a1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E5:27"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="wellfnd1.html#E25:27"><i><font color="Green" title="E29">A28</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c8">a1</font>
<b>by </b><i><a class="txt" href="wellfnd1.html#E25:27"><i><font color="Green" title="E29">A28</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E28:27"/>
contradiction
 <b>by </b><i><a class="txt" href="wellfnd1.html#E2:27"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="wellfnd1.html#E19:27"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="wellfnd1.html#E20:27"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="wellfnd1.html#E21:27"><i><font color="Green" title="E28">A27</font></i></a>, <a class="txt" href="wellfnd1.html#E26:27"><i><font color="Green" title="E30">A29</font></i></a></i>;<br/>


</div>
