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

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon">cR</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">IR</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">CR</font> =  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">ICR</font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">cCR</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:52"/>
not  <a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font> is <a href="neckla_3.html#V1" target="_self">path-connected</a>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; not the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="txt" href="neckla_3.html#E3:52"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">A1</font> =  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> =  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<b>set </b><font color="Maroon">G1</font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font>;<br/>
<b>set </b><font color="Maroon">G2</font> =  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font>;<br/>
<a NAME="E7:52"/><i><font color="Green">E34</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E8:52"/><b>then </b><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E9:52"/>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K6">~</a> 
 
<b>by </b><i><a class="ref" href="relat_2.html#T30">RELAT_2:30</a></i>;<br/>
<a NAME="E10:52"/><b>then </b>
not the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:52"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E11:52"/><b>then </b>
not <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rewrite1.html#D4">REWRITE1:def 4</a></i>;<br/>
<a NAME="E12:52"/><b>then </b>
not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T41">MSUALG_6:41</a></i>;<br/>
<a NAME="E13:52"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:52"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/>
<a NAME="E14:52"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E15:52"/><b>then </b><i><font color="Green">E36</font></i>: 
(  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font> is  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  &amp;  <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font> is  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a>  )
 
<b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E16:52"/><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>

<b>assume </b><a NAME="E2:52_1_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 ;<br/>

<a NAME="E3:52_1_1"/><b>then </b>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:52_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E15:52"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:52_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E15:52"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1"><b>suppose </b></a><a NAME="E1:52_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_2"><b>suppose </b></a><a NAME="E1:52_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E17:52"/><b>then </b><i><font color="Green">E43</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">IG1</font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">IG2</font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">G1G2</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">G2G1</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E18:52"/><i><font color="Green">E60</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#R1">misses</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_2"/>
not the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#R1">misses</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
 ;<br/>

<a NAME="E2:52_2"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:52_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:52_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">c2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:52_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 <b>by </b><i>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/>
<b>consider </b><font color="Maroon">g1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">g2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:52_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, <a class="txt" href="neckla_3.html#E16:52"><i><font color="Green">E39</font></i></a>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/>
<a NAME="E10:52_2"/>
( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E11:52_2"/><b>then </b>
<font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:52_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:52"/><i><font color="Green">E70</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_3"/>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_3_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">g1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">g2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_3_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g1</font>, <font color="Maroon">g2</font> = <font color="Maroon">g2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E5:52_3_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">g1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E16:52"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">g1</font>, <font color="Maroon">u2</font> = <font color="Maroon">g2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E7:52_3_1"/>
<font color="Maroon">u1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">u2</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/>
<a NAME="E8:52_3_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E9:52_3_1"/><i><font color="Green">E105</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_3_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>
 ;<br/>

<a NAME="E2:52_3_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K9">/\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E16:52"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:52_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:52_3_1"/><i><font color="Green">E106</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_3_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:52_3_1_2"/><b>then </b>
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:52_3_1_2"/><b>then </b>
<font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:52_3_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:52_3_1"/><i><font color="Green">E107</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_3_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:52_3_1_3"/><b>then </b>
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:52_3_1_3"/><b>then </b>
<font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:52_3_1_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:52_3_1"/>
<font color="Maroon">a</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="subset_1.html#K7">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E13:52_3_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="neckla_3.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:52_3_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:52_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E3:52_3"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:52_3"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">a</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="subset_1.html#K7">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:52_3"/><b>then </b><i><font color="Green">E110</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">c2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:52_3"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c1</font>,<font color="Maroon">c2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c1</font> = <font color="Maroon">c1</font>, <font color="Maroon">c2</font> = <font color="Maroon">c2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E9:52_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c2</font>
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_3_2"/>
( ( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> ) or ( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> ) or ( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> ) or ( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> ) )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E8:52"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_2_1"><b>suppose </b></a><a NAME="E1:52_3_2_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">c1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon">d2</font> = <font color="Maroon">c2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <b>by </b><i><a class="ref" href="neckla_3.html#T3" target="_self">Th3</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E4:52_3_2_1"/>
<font color="Maroon">d1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d2</font>
 <b>by </b><i>, <a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/><b>hence </b><a NAME="E5:52_3_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_2_2"><b>suppose </b></a><a NAME="E1:52_3_2_2"/>
( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_3_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_2_3"><b>suppose </b></a><a NAME="E1:52_3_2_3"/>
( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_3_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="neckla_3.html#E16:52"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_3_2_4"><b>suppose </b></a><a NAME="E1:52_3_2_4"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">d1</font> = <font color="Maroon">c1</font>, <font color="Maroon">d2</font> = <font color="Maroon">c2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E3:52_3_2_4"/>
<font color="Maroon">d1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">d2</font>
 <b>by </b><i>, <a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, <a class="ref" href="neckla_3.html#T3" target="_self">Th3</a>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/><b>hence </b><a NAME="E4:52_3_2_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E20:52"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">G1G2</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">G2G1</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>thus </b><a NAME="E1:52_4"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_4_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 ;<br/>

<b>assume </b><a NAME="E2:52_4_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E3:52_4_1"/><b>then </b>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:52_4_1"/><b>then </b>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:52_4_1"/><b>then </b><i><font color="Green">E116</font></i>: 
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:52_4_1"/><b>then </b>
(  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="subset_1.html#K7">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:52_4_1"/><b>then </b>
(  not <font color="Maroon">a</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="subset_1.html#K7">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:52_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E16:52"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:52_4"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E3:52_4"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E16:52"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:52_4"/><b>then </b><i><font color="Green">E118</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_4_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_4_2_1"><b>suppose </b></a><a NAME="E1:52_4_2_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_4_2_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 <b>by </b><i>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font>, <font color="Maroon">w</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E5:52_4_2_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">v</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">w</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">v</font>, <font color="Maroon">u2</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><a NAME="E7:52_4_2_1"/>
<font color="Maroon">u1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">u2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/><b>hence </b><a NAME="E8:52_4_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_4_2_2"><b>suppose </b></a><a NAME="E1:52_4_2_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_4_2_2"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, <a class="ref" href="relset_1.html#T6">RELSET_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font>, <font color="Maroon">w</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E5:52_4_2_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">v</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">w</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">v</font>, <font color="Maroon">u2</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><a NAME="E7:52_4_2_2"/>
<font color="Maroon">u1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">u2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/><b>hence </b><a NAME="E8:52_4_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_4_2_3"><b>suppose </b></a><a NAME="E1:52_4_2_3"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_4_2_3"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><a NAME="E4:52_4_2_3"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font>, <font color="Maroon">w</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><a NAME="E6:52_4_2_3"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_4_2_3_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 ;<br/>

<a NAME="E2:52_4_2_3_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:52_4_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E7:52_4_2_3"/><i><font color="Green">E130</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 ;<br/><a NAME="E8:52_4_2_3"/><i><font color="Green">E131</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="neckla_3.html#T3" target="_self">Th3</a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/><a NAME="E9:52_4_2_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:52_4_2_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E11:52_4_2_3"/><b>then </b>
<font color="Maroon">a</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="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E12:52_4_2_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E13:52_4_2_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><a NAME="E14:52_4_2_3"/><b>then </b>
<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="rewrite1.html#T30">REWRITE1:30</a></i>;<br/><a NAME="E15:52_4_2_3"/><b>then </b><i><font color="Green">E132</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="msualg_6.html#T41">MSUALG_6:41</a></i>;<br/><a NAME="E16:52_4_2_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, </i>;<br/><a NAME="E17:52_4_2_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/><a NAME="E18:52_4_2_3"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E19:52_4_2_3"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E20:52_4_2_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_4_2_4"><b>suppose </b></a><a NAME="E1:52_4_2_4"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_4_2_4"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/><a NAME="E4:52_4_2_4"/><i><font color="Green">E173</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font>, <font color="Maroon">w</font> = <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><a NAME="E6:52_4_2_4"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_4_2_4_1"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 ;<br/>

<a NAME="E2:52_4_2_4_1"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:52_4_2_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E7:52_4_2_4"/><i><font color="Green">E176</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 ;<br/><a NAME="E8:52_4_2_4"/><i><font color="Green">E177</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="neckla_3.html#T3" target="_self">Th3</a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/><a NAME="E9:52_4_2_4"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:52_4_2_4"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="subset_1.html#K6">\</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E11:52_4_2_4"/><b>then </b>
<font color="Maroon">a</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="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E12:52_4_2_4"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E13:52_4_2_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="necklace.html#D9">NECKLACE:def 9</a></i>;<br/><a NAME="E14:52_4_2_4"/><b>then </b>
<font color="Maroon">v</font>,<font color="Maroon">w</font> <a href="rewrite1.html#R2">are_convertible_wrt</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="rewrite1.html#T30">REWRITE1:30</a></i>;<br/><a NAME="E15:52_4_2_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="msualg_6.html#T41">MSUALG_6:41</a></i>;<br/><a NAME="E16:52_4_2_4"/><b>then </b><i><font color="Green">E178</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/><a NAME="E17:52_4_2_4"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, </i>;<br/><a NAME="E18:52_4_2_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="msualg_5.html#K1">EqCl</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3">ComplRelStr</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/><a NAME="E19:52_4_2_4"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="neckla_3.html#K1" target="_self">component</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E20:52_4_2_4"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:52"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E21:52_4_2_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E21:52"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K2">sum_of</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A1</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">A2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="neckla_2.html#D3">NECKLA_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E22:52"/>
 ex <font color="Olive">G1</font>, <font color="Olive">G2</font> being non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a> <a href="necklace.html#V1">symmetric</a> <a href="necklace.html#V3">irreflexive</a>  <a href="orders_2.html#L1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G1</font> <a href="subset_1.html#R1">misses</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G2</font> &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> #) <a href="hidden.html#R1">=</a>  <a href="neckla_2.html#K2">sum_of</a> <font color="Olive">G1</font>,<font color="Olive">G2</font> )
 <b>by </b><i>, <a class="txt" href="neckla_3.html#E7:52"><i><font color="Green">E34</font></i></a>, <a class="txt" href="neckla_3.html#E15:52"><i><font color="Green">E36</font></i></a></i>;<br/>


</div>
