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

<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">cR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c3">IR</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>assume </b><a NAME="E1:51"/>
not <font color="Maroon" title="c1">R</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:51"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font> &amp; not the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="neckla_3.html#D2" target="_self" title="NECKLA_3:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E3:51"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">A1</font> =  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>;<br/>
<b>set </b><font color="Maroon" title="c9">A2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">A2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>set </b><font color="Maroon" title="c11">G1</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c12">G2</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font>;<br/>
<a NAME="E6:51"/><i><font color="Green" title="E31">A2</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A2</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E7:51"/><b>then </b><i><font color="Green" title="E32">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E8:51"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> 
 
<b>by </b><i><a class="ref" href="relat_2.html#T30" title="RELAT_2:th.30">RELAT_2:30</a></i>;<br/>
<a NAME="E9:51"/><b>then </b>
not the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:51"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E10:51"/><b>then </b>
not <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#D4" title="REWRITE1:def.4">REWRITE1:def 4</a></i>;<br/>
<a NAME="E11:51"/><b>then </b>
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="msualg_6.html#T41" title="MSUALG_6:th.41">MSUALG_6:41</a></i>;<br/>
<a NAME="E12:51"/><b>then </b>
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> )
 
<b>by </b><i><a class="ref" href="neckla_3.html#T30" target="_self" title="NECKLA_3:th.30">Th30</a></i>;<br/>
<a NAME="E13:51"/><b>then </b>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E14:51"/><b>then </b><i><font color="Green" title="E33">A4</font></i>: 
(  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> is  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  &amp;  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font> is  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E15:51"/><i><font color="Green" title="E34">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">A2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:51_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">A2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_1_1"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ;<br/>

<b>assume </b><a NAME="E2:51_1_1"/>
not <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">A2</font>
 ;<br/>

<a NAME="E3:51_1_1"/><b>then </b>
( not <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; not <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:51_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:51_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:51_1"/>
<span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:51"/><b>then </b><i><font color="Green" title="E35">A7</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p3">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/>
<a NAME="E17:51"/><i><font color="Green" title="E36">A8</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c13">IG1</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">IG2</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:51_2"/>
not the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 ;<br/>

<a NAME="E2:51_2"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:51_2"/><i><font color="Green" title="E37">A9</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:51_2"/><i><font color="Green" title="E38">A10</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E4:51_2"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">c1</font>, <font color="Maroon" title="c17">c2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:51_2"/><i><font color="Green" title="E39">A11</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c17">c2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c16">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c17">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">g1</font>, <font color="Maroon" title="c19">g2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:51_2"/><i><font color="Green" title="E40">A12</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">g1</font>,<font color="Maroon" title="c19">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c18">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c19">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="neckla_3.html#E5:51_2"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/>
<a NAME="E10:51_2"/>
( <font color="Maroon" title="c16">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c16">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E7:51_2"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="neckla_3.html#E9:51_2"><i><font color="Green" title="E40">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E11:51_2"/><b>then </b>
<font color="Maroon" title="c16">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">A2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:51_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="neckla_3.html#E7:51"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:51"/><i><font color="Green" title="E37">A13</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c13">IG1</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">IG2</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>;<br/>
<b>thus </b><a NAME="E1:51_3"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_3_1"/><i><font color="Green" title="E38">A14</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">g1</font>, <font color="Maroon" title="c17">g2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:51_3_1"/><i><font color="Green" title="E39">A15</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c17">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c17">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">g1</font> = <font color="Maroon" title="c16">g1</font>, <font color="Maroon" title="c19">g2</font> = <font color="Maroon" title="c17">g2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_3_1"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E5:51_3_1"/><i><font color="Green" title="E40">A16</font></i>: 
<font color="Maroon" title="c18">g1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">g2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:51_3_1"><i><font color="Green" title="E38">A14</font></i></a>, <a class="txt" href="neckla_3.html#E3:51_3_1"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">u1</font> = <font color="Maroon" title="c18">g1</font>, <font color="Maroon" title="c21">u2</font> = <font color="Maroon" title="c19">g2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_3_1"><i><font color="Green" title="E39">A15</font></i></a></i>;<br/>
<a NAME="E7:51_3_1"/>
<font color="Maroon" title="c20">u1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">u2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:51_3_1"><i><font color="Green" title="E40">A16</font></i></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/>
<a NAME="E8:51_3_1"/><b>then </b><i><font color="Green" title="E41">A17</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:51_3_1"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E9:51_3_1"/>
not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_3_1_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:51_3_1_1"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:51_3_1"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:51_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:51"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:51_3_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:51_3_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c15">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:51_3"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:51_3"/><b>then </b><i><font color="Green" title="E38">A18</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> &amp; not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">c1</font>, <font color="Maroon" title="c17">c2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:51_3"/><i><font color="Green" title="E39">A19</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c17">c2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c16">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c17">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">c1</font> = <font color="Maroon" title="c16">c1</font>, <font color="Maroon" title="c19">c2</font> = <font color="Maroon" title="c17">c2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E5:51_3"><i><font color="Green" title="E39">A19</font></i></a></i>;<br/>
<a NAME="E7:51_3"/><i><font color="Green" title="E40">A20</font></i>: 
<font color="Maroon" title="c18">c1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">c2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:51_3"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="neckla_3.html#E5:51_3"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_3_2"/>
( ( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> ) or ( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> ) or ( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> ) or ( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> ) )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E15:51"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2_1"><b>suppose </b></a><a NAME="E1:51_3_2_1"/><i><font color="Green" title="E41">A21</font></i>: 
( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c20">d1</font> = <font color="Maroon" title="c18">c1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">d2</font> = <font color="Maroon" title="c19">c2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="neckla_3.html#E1:51_3_2_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><a NAME="E4:51_3_2_1"/>
<font color="Maroon" title="c20">d1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">d2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="neckla_3.html#E7:51_3"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>;<br/><b>hence </b><a NAME="E5:51_3_2_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_3"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="neckla_3.html#E5:51_3"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2_2"><b>suppose </b></a><a NAME="E1:51_3_2_2"/>
( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 ;<br/><div class="add"><a NAME="E2:51_3_2_2"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:51_3"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E3:51_3_2_2"/><b>then </b><i><font color="Green" title="E41">A22</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_3"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:51_3_2_2"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_3_2_2_1"/>
not <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 ;<br/>

<a NAME="E2:51_3_2_2_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:51_3_2_2_1"/><i><font color="Green" title="E42">A23</font></i>: 
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:51_3_2_2_1"/><i><font color="Green" title="E43">A24</font></i>: 
( <font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E4:51_3_2_2_1"><i><font color="Green" title="E42">A23</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">b1</font>, <font color="Maroon" title="c22">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:51_3_2_2_1"/><i><font color="Green" title="E44">A25</font></i>: 
( <font color="Maroon" title="c21">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c22">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">b1</font>,<font color="Maroon" title="c22">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">b1</font> = <font color="Maroon" title="c21">b1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E7:51_3_2_2_1"><i><font color="Green" title="E44">A25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">b2</font> = <font color="Maroon" title="c22">b2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E7:51_3_2_2_1"><i><font color="Green" title="E44">A25</font></i></a></i>;<br/>
<a NAME="E10:51_3_2_2_1"/><i><font color="Green" title="E45">A26</font></i>: 
( <font color="Maroon" title="c23">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c24">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 
<b>by </b><i><a class="ref" href="msualg_5.html#D1" title="MSUALG_5:def.1">MSUALG_5:def 1</a></i>;<br/>
<a NAME="E11:51_3_2_2_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c23">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E7:51_3_2_2_1"><i><font color="Green" title="E44">A25</font></i></a>, <a class="ref" href="neckla_3.html#T30" target="_self" title="NECKLA_3:th.30">Th30</a></i>;<br/>
<a NAME="E12:51_3_2_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c24">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:51_3_2_2_1"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="neckla_3.html#E7:51_3_2_2_1"><i><font color="Green" title="E44">A25</font></i></a>, <a class="txt" href="neckla_3.html#E10:51_3_2_2_1"><i><font color="Green" title="E45">A26</font></i></a>, <a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></i>;<br/>
<a NAME="E13:51_3_2_2_1"/><b>then </b>
<font color="Maroon" title="c24">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="ref" href="neckla_3.html#T30" target="_self" title="NECKLA_3:th.30">Th30</a></i>;<br/>
<a NAME="E14:51_3_2_2_1"/><b>then </b>
<font color="Maroon" title="c24">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">A2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E7:51_3_2_2_1"><i><font color="Green" title="E44">A25</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E15:51_3_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="neckla_3.html#E7:51"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:51_3_2_2"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_3_2_2"><i><font color="Green" title="E41">A22</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2_3"><b>suppose </b></a><a NAME="E1:51_3_2_3"/>
( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> )
 ;<br/><div class="add"><a NAME="E2:51_3_2_3"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:51_3"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E3:51_3_2_3"/><b>then </b><i><font color="Green" title="E41">A27</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_3"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:51_3_2_3"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_3_2_3_1"/>
not <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 ;<br/>

<a NAME="E2:51_3_2_3_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:51_3_2_3_1"/><i><font color="Green" title="E42">A28</font></i>: 
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:51_3_2_3_1"/><i><font color="Green" title="E43">A29</font></i>: 
( <font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E4:51_3_2_3_1"><i><font color="Green" title="E42">A28</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">b1</font>, <font color="Maroon" title="c22">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:51_3_2_3_1"/><i><font color="Green" title="E44">A30</font></i>: 
( <font color="Maroon" title="c21">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c22">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">b1</font>,<font color="Maroon" title="c22">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">b1</font> = <font color="Maroon" title="c21">b1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E7:51_3_2_3_1"><i><font color="Green" title="E44">A30</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">b2</font> = <font color="Maroon" title="c22">b2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E7:51_3_2_3_1"><i><font color="Green" title="E44">A30</font></i></a></i>;<br/>
<a NAME="E10:51_3_2_3_1"/>
( <font color="Maroon" title="c23">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c24">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 
<b>by </b><i><a class="ref" href="msualg_5.html#D1" title="MSUALG_5:def.1">MSUALG_5:def 1</a></i>;<br/>
<a NAME="E11:51_3_2_3_1"/><b>then </b><i><font color="Green" title="E45">A31</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">b2</font>,<font color="Maroon" title="c23">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:51_3_2_3_1"><i><font color="Green" title="E43">A29</font></i></a>, <a class="txt" href="neckla_3.html#E7:51_3_2_3_1"><i><font color="Green" title="E44">A30</font></i></a>, <a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a></i>;<br/>
<a NAME="E12:51_3_2_3_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c24">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E7:51_3_2_3_1"><i><font color="Green" title="E44">A30</font></i></a>, <a class="ref" href="neckla_3.html#T30" target="_self" title="NECKLA_3:th.30">Th30</a></i>;<br/>
<a NAME="E13:51_3_2_3_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c23">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E11:51_3_2_3_1"><i><font color="Green" title="E45">A31</font></i></a>, <a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></i>;<br/>
<a NAME="E14:51_3_2_3_1"/><b>then </b>
<font color="Maroon" title="c23">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="ref" href="neckla_3.html#T30" target="_self" title="NECKLA_3:th.30">Th30</a></i>;<br/>
<a NAME="E15:51_3_2_3_1"/><b>then </b>
<font color="Maroon" title="c23">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">A2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E7:51_3_2_3_1"><i><font color="Green" title="E44">A30</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E16:51_3_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="neckla_3.html#E7:51"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:51_3_2_3"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_3_2_3"><i><font color="Green" title="E41">A27</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2_4"><b>suppose </b></a><a NAME="E1:51_3_2_4"/><i><font color="Green" title="E41">A32</font></i>: 
( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c20">d1</font> = <font color="Maroon" title="c18">c1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">d2</font> = <font color="Maroon" title="c19">c2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <b>by </b><i><a class="txt" href="neckla_3.html#E1:51_3_2_4"><i><font color="Green" title="E41">A32</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><a NAME="E4:51_3_2_4"/>
<font color="Maroon" title="c20">d1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">d2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="neckla_3.html#E7:51_3"><i><font color="Green" title="E40">A20</font></i></a>, <a class="txt" href="neckla_3.html#E1:51_3_2_4"><i><font color="Green" title="E41">A32</font></i></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>;<br/><b>hence </b><a NAME="E5:51_3_2_4"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:51_3"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E19:51"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c13">IG1</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">IG2</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>;<br/>
<b>thus </b><a NAME="E1:51_4"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_4_1"/><i><font color="Green" title="E38">A33</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 ;<br/>

<b>assume </b><a NAME="E2:51_4_1"/>
not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 ;<br/>

<a NAME="E3:51_4_1"/><b>then </b>
( not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> &amp; not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:51_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E18:51"><i><font color="Green" title="E37">A13</font></i></a>, <a class="txt" href="neckla_3.html#E1:51_4_1"><i><font color="Green" title="E38">A33</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c15">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:51_4"/><i><font color="Green" title="E38">A34</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_4_2"/>
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> or <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E2:51_4"><i><font color="Green" title="E38">A34</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_2_1"><b>suppose </b></a><a NAME="E1:51_4_2_1"/><i><font color="Green" title="E39">A35</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c17">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:51_4_2_1"/><i><font color="Green" title="E40">A36</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v</font>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">v</font> = <font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c19">w</font> = <font color="Maroon" title="c17">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_4_2_1"><i><font color="Green" title="E40">A36</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><a NAME="E5:51_4_2_1"/><i><font color="Green" title="E41">A37</font></i>: 
<font color="Maroon" title="c18">v</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">w</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:51_4_2_1"><i><font color="Green" title="E39">A35</font></i></a>, <a class="txt" href="neckla_3.html#E3:51_4_2_1"><i><font color="Green" title="E40">A36</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">u1</font> = <font color="Maroon" title="c18">v</font>, <font color="Maroon" title="c21">u2</font> = <font color="Maroon" title="c19">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_4_2_1"><i><font color="Green" title="E40">A36</font></i></a></i>;<br/><a NAME="E7:51_4_2_1"/>
<font color="Maroon" title="c20">u1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">u2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:51_4_2_1"><i><font color="Green" title="E41">A37</font></i></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/><b>hence </b><a NAME="E8:51_4_2_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_4_2_1"><i><font color="Green" title="E40">A36</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_2_2"><b>suppose </b></a><a NAME="E1:51_4_2_2"/><i><font color="Green" title="E39">A38</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c17">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:51_4_2_2"/><i><font color="Green" title="E40">A39</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v</font>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E6:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">v</font> = <font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c19">w</font> = <font color="Maroon" title="c17">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_4_2_2"><i><font color="Green" title="E40">A39</font></i></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/><a NAME="E5:51_4_2_2"/><i><font color="Green" title="E41">A40</font></i>: 
<font color="Maroon" title="c18">v</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">w</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:51_4_2_2"><i><font color="Green" title="E39">A38</font></i></a>, <a class="txt" href="neckla_3.html#E3:51_4_2_2"><i><font color="Green" title="E40">A39</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">u1</font> = <font color="Maroon" title="c18">v</font>, <font color="Maroon" title="c21">u2</font> = <font color="Maroon" title="c19">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_4_2_2"><i><font color="Green" title="E40">A39</font></i></a></i>;<br/><a NAME="E7:51_4_2_2"/>
<font color="Maroon" title="c20">u1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">u2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:51_4_2_2"><i><font color="Green" title="E41">A40</font></i></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/><b>hence </b><a NAME="E8:51_4_2_2"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:51_4_2_2"><i><font color="Green" title="E40">A39</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E20:51"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p3">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E21:51"/>
 ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a> <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G2</font> &amp;  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:51"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="neckla_3.html#E14:51"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="neckla_3.html#E16:51"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/>


</div>
