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

<b>let </b><font color="Maroon" title="c1">G</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>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">R1</font>, <font color="Maroon" title="c4">R2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green" title="E37">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">R1</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="Maroon" title="c4">R2</font>
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green" title="E38">A2</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <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="Maroon" title="c3">R1</font>,<font color="Maroon" title="c4">R2</font>
 <b>and </b><br/><a NAME="E3:61"/><i><font color="Green" title="E39">A3</font></i>: 
( not <font color="Maroon" title="c1">G</font> is <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a> &amp; <font color="Maroon" title="c1">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a> &amp;  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a> )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c6">X</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">X</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<b>set </b><font color="Maroon" title="c8">H</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">X</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<b>set </b><font color="Maroon" title="c10">R</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c9">A</font>;<br/>
<a NAME="E6:61"/>
not  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c9">A</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c9">A</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 ;<br/>

<a NAME="E2:61_1"/><b>then </b>
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="c9">A</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E3:61_1"/><b>then </b>
<font color="Maroon" title="c9">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<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="E4:61_1"/><b>then </b><i><font color="Green" title="E40">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T66" title="ZFMISC_1:th.66">ZFMISC_1:66</a></i>;<br/>
<a NAME="E5:61_1"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:61"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="realset2.html#D5" title="REALSET2:def.5">REALSET2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:61_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:61_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">R</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c9">A</font> as  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/>
<a NAME="E8:61"/><i><font color="Green" title="E40">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</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 <font color="Maroon" title="c11">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:61_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</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="c11">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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="61_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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="E1:61_2_1"/><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c12">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">G</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_2_1_1"/>
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_2_1_1_1"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:61_2_1_1_1"/>
<font color="Maroon" title="c12">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="c11">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_2_1_1_2"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 ;<br/><div class="add"><a NAME="E2:61_2_1_1_2"/><b>then </b>
not <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:61_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">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">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_2_1"><i><font color="Green" title="E41">A6</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="E4:61_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">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="c11">R</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/><b>hence </b><a NAME="E5:61_2_1_1_2"/>
<font color="Maroon" title="c12">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="c11">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon" title="c12">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="E2:61_2"/><i><font color="Green" title="E41">A7</font></i>: 
<font color="Maroon" title="c12">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="c11">R</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_2_2"/>
( <font color="Maroon" title="c12">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="c11">R</font> or <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E2:61_2"><i><font color="Green" title="E41">A7</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="61_2_2_1"><b>suppose </b></a><a NAME="E1:61_2_2_1"/>
<font color="Maroon" title="c12">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="c11">R</font>
 ;<br/><div class="add"><a NAME="E2:61_2_2_1"/><b>then </b>
<font color="Maroon" title="c12">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">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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>hence </b><a NAME="E3:61_2_2_1"/>
<font color="Maroon" title="c12">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">G</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_2_2_2"><b>suppose </b></a><a NAME="E1:61_2_2_2"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_2_2_2"/>
<font color="Maroon" title="c12">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">G</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:61"/><i><font color="Green" title="E41">A8</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> <font color="Maroon" title="c7">X</font></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 <font color="Maroon" title="c11">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_3"/>
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="c7">X</font></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 <font color="Maroon" title="c11">R</font>
 ;<br/>

<a NAME="E2:61_3"/><b>then </b>
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="c7">X</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">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/>
<a NAME="E3:61_3"/><b>then </b>
<font color="Maroon" title="c7">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">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="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E4:61_3"/><b>then </b>
<font color="Maroon" title="c7">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">A</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="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:61_3"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">A</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="E7:61_3"/>
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E6:61_3"><i><font color="Green" title="E42">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>hence </b><a NAME="E8:61_3"/>
contradiction
 <b>by </b><i><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>
<a NAME="E10:61"/>
not  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">X</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_4"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">X</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 ;<br/>

<a NAME="E2:61_4"/><b>then </b>
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="c7">X</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:61_4"/>
contradiction
 <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">H</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">X</font> as  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/>
<a NAME="E12:61"/><i><font color="Green" title="E42">A10</font></i>: 
<font color="Maroon" title="c11">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="Maroon" title="c3">R1</font>,<font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E2:61"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">X1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font> : <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> </span>}</span> , <font color="Maroon" title="c14">Y1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font> : not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> </span>}</span> , <font color="Maroon" title="c15">X2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font> : <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> </span>}</span> , <font color="Maroon" title="c16">Y2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font> : not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> </span>}</span>  as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E14:61"/><i><font color="Green" title="E43">A11</font></i>: 
<font color="Maroon" title="c13">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c14">Y1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_5"/>
not <font color="Maroon" title="c13">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c14">Y1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:61_5"/><i><font color="Green" title="E44">A12</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E5:61_5"/><i><font color="Green" title="E45">A13</font></i>: 
( <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_5"><i><font color="Green" title="E44">A12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E7:61_5"/><i><font color="Green" title="E46">A14</font></i>: 
( <font color="Maroon" title="c19">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_5"><i><font color="Green" title="E44">A12</font></i></a></i>;<br/>
<b>thus </b><a NAME="E8:61_5"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:61_5"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="neckla_3.html#E7:61_5"><i><font color="Green" title="E46">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:61"/><i><font color="Green" title="E44">A15</font></i>: 
<font color="Maroon" title="c15">X2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c16">Y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_6"/>
not <font color="Maroon" title="c15">X2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c16">Y2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:61_6"/><i><font color="Green" title="E45">A16</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E5:61_6"/><i><font color="Green" title="E46">A17</font></i>: 
( <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_6"><i><font color="Green" title="E45">A16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E7:61_6"/><i><font color="Green" title="E47">A18</font></i>: 
( <font color="Maroon" title="c19">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_6"><i><font color="Green" title="E45">A16</font></i></a></i>;<br/>
<b>thus </b><a NAME="E8:61_6"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:61_6"><i><font color="Green" title="E46">A17</font></i></a>, <a class="txt" href="neckla_3.html#E7:61_6"><i><font color="Green" title="E47">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:61"/><i><font color="Green" title="E45">A19</font></i>: 
<font color="Maroon" title="c11">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
;<br/>
<a NAME="E17:61"/><i><font color="Green" title="E46">A20</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">Y1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:61_7"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">R1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">Y1</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="61_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">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="E1:61_7_1"/><i><font color="Green" title="E47">A21</font></i>: 
<font color="Maroon" title="c17">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="c3">R1</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_7_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_7_1_1_1"><b>suppose </b></a><a NAME="E1:61_7_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 ;<br/><div class="add"><a NAME="E2:61_7_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_7_1"><i><font color="Green" title="E47">A21</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_7_1_1_1"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">Y1</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_7_1_1_2"><b>suppose </b></a><a NAME="E1:61_7_1_1_2"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 ;<br/><div class="add"><a NAME="E2:61_7_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_7_1"><i><font color="Green" title="E47">A21</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_7_1_1_2"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">Y1</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/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon" title="c17">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="E2:61_7"/><i><font color="Green" title="E47">A22</font></i>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">Y1</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_7_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_7_2"/>
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font> or <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E2:61_7"><i><font color="Green" title="E47">A22</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="61_7_2_1"><b>suppose </b></a><a NAME="E1:61_7_2_1"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_7_2_1"/><i><font color="Green" title="E48">A23</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/><b>thus </b><a NAME="E4:61_7_2_1"/>
<font color="Maroon" title="c17">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="c3">R1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_7_2_1"><i><font color="Green" title="E48">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_7_2_2"><b>suppose </b></a><a NAME="E1:61_7_2_2"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_7_2_2"/><i><font color="Green" title="E48">A24</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/><b>thus </b><a NAME="E4:61_7_2_2"/>
<font color="Maroon" title="c17">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="c3">R1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_7_2_2"><i><font color="Green" title="E48">A24</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E18:61"/><i><font color="Green" title="E47">A25</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">X2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">Y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_8"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:61_8"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">X2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">Y2</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="61_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">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="E1:61_8_1"/><i><font color="Green" title="E48">A26</font></i>: 
<font color="Maroon" title="c17">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="c4">R2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_8_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_8_1_1_1"><b>suppose </b></a><a NAME="E1:61_8_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 ;<br/><div class="add"><a NAME="E2:61_8_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_8_1"><i><font color="Green" title="E48">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_8_1_1_1"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">Y2</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_8_1_1_2"><b>suppose </b></a><a NAME="E1:61_8_1_1_2"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 ;<br/><div class="add"><a NAME="E2:61_8_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_8_1"><i><font color="Green" title="E48">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E3:61_8_1_1_2"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">Y2</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/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>let </b><font color="Maroon" title="c17">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="E2:61_8"/><i><font color="Green" title="E48">A27</font></i>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">Y2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_8_2"/>
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font> or <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E2:61_8"><i><font color="Green" title="E48">A27</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="61_8_2_1"><b>suppose </b></a><a NAME="E1:61_8_2_1"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_8_2_1"/><i><font color="Green" title="E49">A28</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/><b>thus </b><a NAME="E4:61_8_2_1"/>
<font color="Maroon" title="c17">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="c4">R2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_8_2_1"><i><font color="Green" title="E49">A28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_8_2_2"><b>suppose </b></a><a NAME="E1:61_8_2_2"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_8_2_2"/><i><font color="Green" title="E49">A29</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/><b>thus </b><a NAME="E4:61_8_2_2"/>
<font color="Maroon" title="c17">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="c4">R2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_8_2_2"><i><font color="Green" title="E49">A29</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E19:61"/><i><font color="Green" title="E48">A30</font></i>: 
<font color="Maroon" title="c11">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="Maroon" title="c4">R2</font>,<font color="Maroon" title="c3">R1</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_3.html#T8" target="_self" title="NECKLA_3:th.8">Th8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E21:61"/><i><font color="Green" title="E49">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="c17">a</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="neckla_3.html#E2:61"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="neckla_3.html#E3:61"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="neckla_3.html#T38" target="_self" title="NECKLA_3:th.38">Th38</a></i>;<br/>
<a NAME="E22:61"/><i><font color="Green" title="E50">A32</font></i>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E21:61"><i><font color="Green" title="E49">A31</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E24:61"/><i><font color="Green" title="E51">A33</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c18">b</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="neckla_3.html#E3:61"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="neckla_3.html#E16:61"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="neckla_3.html#E19:61"><i><font color="Green" title="E48">A30</font></i></a>, <a class="ref" href="neckla_3.html#T38" target="_self" title="NECKLA_3:th.38">Th38</a></i>;<br/>
<a NAME="E25:61"/><i><font color="Green" title="E52">A34</font></i>: 
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E24:61"><i><font color="Green" title="E51">A33</font></i></a></i>;<br/>
<a NAME="E26:61"/><i><font color="Green" title="E53">A35</font></i>: 
not <font color="Maroon" title="c14">Y1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">Y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_9"/>
<font color="Maroon" title="c14">Y1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">Y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<a NAME="E2:61_9"/><b>then </b><i><font color="Green" title="E54">A36</font></i>: 
( <font color="Maroon" title="c14">Y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c16">Y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E3:61_9"/><i><font color="Green" title="E55">A37</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">R</font> holds  <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">R</font>;<br/>

<a NAME="E1:61_9_1"/><i><font color="Green" title="E56">A38</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">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 <font color="Maroon" title="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_9_1_1"/>
( <font color="Maroon" title="c19">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="c3">R1</font> or <font color="Maroon" title="c19">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="c4">R2</font> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E1:61_9_1"><i><font color="Green" title="E56">A38</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="61_9_1_1_1"><b>suppose </b></a><a NAME="E1:61_9_1_1_1"/>
<font color="Maroon" title="c19">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="c3">R1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c20">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_9_1_1_1"/><i><font color="Green" title="E57">A39</font></i>: 
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:61"><i><font color="Green" title="E46">A20</font></i></a>, <a class="txt" href="neckla_3.html#E2:61_9"><i><font color="Green" title="E54">A36</font></i></a></i>;<br/><b>thus </b><a NAME="E4:61_9_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_9_1_1_1"><i><font color="Green" title="E57">A39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_9_1_1_2"><b>suppose </b></a><a NAME="E1:61_9_1_1_2"/>
<font color="Maroon" title="c19">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="c4">R2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c20">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_9_1_1_2"/><i><font color="Green" title="E57">A40</font></i>: 
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E18:61"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="neckla_3.html#E2:61_9"><i><font color="Green" title="E54">A36</font></i></a></i>;<br/><b>thus </b><a NAME="E4:61_9_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_9_1_1_2"><i><font color="Green" title="E57">A40</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:61_9"/>
not  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_9_2"/><i><font color="Green" title="E56">A41</font></i>: 
 <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font> is <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a>
 ;<br/>

<a NAME="E2:61_9_2"/><i><font color="Green" title="E57">A42</font></i>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_9_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_9_2_1"/>
not <font color="Maroon" title="c17">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<a NAME="E2:61_9_2_1"/><b>then </b>
<font color="Maroon" title="c2">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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</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/>
<a NAME="E3:61_9_2_1"/><b>then </b><i><font color="Green" title="E58">A43</font></i>: 
<font color="Maroon" title="c2">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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E4:61_9_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:61_9_2_1"/><b>then </b>
<font color="Maroon" title="c2">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="c12">H</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="E6:61_9_2_1"/><b>then </b>
<font color="Maroon" title="c2">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="c11">R</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">H</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:61_9_2_1"><i><font color="Green" title="E58">A43</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="E7:61_9_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:61"><i><font color="Green" title="E41">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>
<a NAME="E3:61_9_2"/>
<font color="Maroon" title="c17">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_9_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:61_9_2_2"/>
<font color="Maroon" title="c17">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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</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/>
<a NAME="E2:61_9_2_2"/><b>then </b><i><font color="Green" title="E58">A44</font></i>: 
<font color="Maroon" title="c17">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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E3:61_9_2_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:61"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:61_9_2_2"/>
<font color="Maroon" title="c17">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:61_9_2_2"><i><font color="Green" title="E58">A44</font></i></a>, <a class="ref" href="necklace.html#D9" title="NECKLACE:def.9">NECKLACE:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:61_9_2"/><b>then </b>
( <font color="Maroon" title="c17">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c2">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="necklace.html#D9" title="NECKLACE:def.9">NECKLACE:def 9</a></i>;<br/>
<a NAME="E5:61_9_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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c17">a</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_9_2"><i><font color="Green" title="E56">A41</font></i></a>, <a class="txt" href="neckla_3.html#E2:61_9_2"><i><font color="Green" title="E57">A42</font></i></a>, <a class="ref" href="neckla_3.html#D2" target="_self" title="NECKLA_3:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E7:61_9_2"/><i><font color="Green" title="E58">A45</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> )
 <b>and </b><br/><a NAME="E8:61_9_2"/><i><font color="Green" title="E59">A46</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">p</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">p</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="rewrite1.html#T12" title="REWRITE1:th.12">REWRITE1:12</a></i>;<br/>
<a NAME="E9:61_9_2"/><i><font color="Green" title="E60">A47</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">p</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E7:61_9_2"><i><font color="Green" title="E58">A45</font></i></a>, <a class="ref" href="nat_1.html#T25" title="NAT_1:th.25">NAT_1:25</a></i>;<br/>
<a NAME="E10:61_9_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="neckla_3.html#E2:61_9_2"><i><font color="Green" title="E57">A42</font></i></a>, <a class="txt" href="neckla_3.html#E7:61_9_2"><i><font color="Green" title="E58">A45</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E11:61_9_2"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T25" title="NAT_1:th.25">NAT_1:25</a></i>;<br/>
<a NAME="E12:61_9_2"/><b>then </b>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">p</font> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">p</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E9:61_9_2"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E13:61_9_2"/><b>then </b><i><font color="Green" title="E61">A48</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:61_9_2"><i><font color="Green" title="E59">A46</font></i></a></i>;<br/>
<a NAME="E14:61_9_2"/><b>then </b>
<font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <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 <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E15:61_9_2"/><b>then </b><i><font color="Green" title="E62">A49</font></i>: 
<font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <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">G</font>
 
<b>by </b><i><a class="ref" href="necklace.html#D9" title="NECKLACE:def.9">NECKLACE:def 9</a></i>;<br/>
<a NAME="E16:61_9_2"/><i><font color="Green" title="E63">A50</font></i>: 
<font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_9_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_9_2_3"/><i><font color="Green" title="E64">A51</font></i>: 
not <font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<a NAME="E2:61_9_2_3"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</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="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E3:61_9_2_3"/><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="c2">x</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E7:61_9_2"><i><font color="Green" title="E58">A45</font></i></a>, <a class="txt" href="neckla_3.html#E13:61_9_2"><i><font color="Green" title="E61">A48</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_9_2_3"><i><font color="Green" title="E64">A51</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="E4:61_9_2_3"/><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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<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>hence </b><a NAME="E5:61_9_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="neckla_3.html#T13" target="_self" title="NECKLA_3:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:61_9_2"/><i><font color="Green" title="E64">A52</font></i>: 
<font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <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="c11">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_9_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_9_2_4"/>
not <font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <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="c11">R</font>
 ;<br/>

<a NAME="E2:61_9_2_4"/><b>then </b>
<font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:61"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="neckla_3.html#E15:61_9_2"><i><font color="Green" title="E62">A49</font></i></a>, <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="E3:61_9_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E16:61_9_2"><i><font color="Green" title="E63">A50</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:61_9_2"/><i><font color="Green" title="E65">A53</font></i>: 
( <font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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 <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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 <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E13:61_9_2"><i><font color="Green" title="E61">A48</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E19:61_9_2"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a></i>;<br/>
<a NAME="E20:61_9_2"/><b>then </b><i><font color="Green" title="E66">A54</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E13:61_9_2"><i><font color="Green" title="E61">A48</font></i></a>, <a class="txt" href="neckla_3.html#E18:61_9_2"><i><font color="Green" title="E65">A53</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/>
<a NAME="E21:61_9_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:61_9"><i><font color="Green" title="E55">A37</font></i></a>, <a class="txt" href="neckla_3.html#E17:61_9_2"><i><font color="Green" title="E64">A52</font></i></a></i>;<br/>
<a NAME="E22:61_9_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></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 <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E7:61_9_2"><i><font color="Green" title="E58">A45</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_9_2"><i><font color="Green" title="E66">A54</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="E23:61_9_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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 
<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>hence </b><a NAME="E24:61_9_2"/>
contradiction
 <b>by </b><i><a class="ref" href="neckla_3.html#T12" target="_self" title="NECKLA_3:th.12">Th12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:61_9"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61"><i><font color="Green" title="E39">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E27:61"/>
<font color="Maroon" title="c1">G</font> <a href="necklace.html#R2" title="NECKLACE:pred.2">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_10_1"/>
(  not <font color="Maroon" title="c14">Y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c16">Y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E26:61"><i><font color="Green" title="E53">A35</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1"><b>suppose </b></a><a NAME="E1:61_10_1_1"/><i><font color="Green" title="E54">A55</font></i>: 
not <font color="Maroon" title="c14">Y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:61_10_1_1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Y1</font> ex <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">X1</font> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c19">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Y1</font>;<br/>
<a NAME="E2:61_10_1_1_1"/>
<font color="Maroon" title="c17">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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</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/>
<a NAME="E3:61_10_1_1_1"/><b>then </b><i><font color="Green" title="E55">A56</font></i>: 
<font color="Maroon" title="c17">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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E4:61_10_1_1_1"/><i><font color="Green" title="E56">A57</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:61"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">a</font> = <font color="Maroon" title="c17">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_10_1_1_1"><i><font color="Green" title="E55">A56</font></i></a></i>;<br/>
<a NAME="E6:61_10_1_1_1"/>
<font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1"><i><font color="Green" title="E54">A55</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E8:61_10_1_1_1"/><i><font color="Green" title="E57">A58</font></i>: 
( <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/>
<a NAME="E9:61_10_1_1_1"/>
<font color="Maroon" title="c19">b</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_1_1"><i><font color="Green" title="E57">A58</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E10:61_10_1_1_1"/><b>then </b>
<font color="Maroon" title="c19">b</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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">b</font> = <font color="Maroon" title="c19">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="neckla_3.html#E4:61_10_1_1_1"><i><font color="Green" title="E56">A57</font></i></a></i>;<br/>
<a NAME="E12:61_10_1_1_1"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_1_1"/>
not <font color="Maroon" title="c20">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">b</font>
 ;<br/>

<a NAME="E2:61_10_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font> &amp; <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E21:61"><i><font color="Green" title="E49">A31</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_1"><i><font color="Green" title="E54">A55</font></i></a></i>;<br/>
<a NAME="E3:61_10_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y1</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="E4:61_10_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E14:61"><i><font color="Green" title="E43">A11</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="E13:61_10_1_1_1"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c20">a</font>,<font color="Maroon" title="c22">b</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:61"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="neckla_3.html#D2" target="_self" title="NECKLA_3:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E15:61_10_1_1_1"/><i><font color="Green" title="E58">A59</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">b</font> )
 <b>and </b><br/><a NAME="E16:61_10_1_1_1"/><i><font color="Green" title="E59">A60</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T12" title="REWRITE1:th.12">REWRITE1:12</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> $1 &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> holds <br/><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> ) );<br/>
<a NAME="E17:61_10_1_1_1"/><i><font color="Green" title="E60">A61</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_1_1_2_1"/><i><font color="Green" title="E61">A62</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_1_1_2_1_1"/>
<font color="Maroon" title="c23">p</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="txt" href="neckla_3.html#E15:61_10_1_1_1"><i><font color="Green" title="E58">A59</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E2:61_10_1_1_1_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_10_1_1_1_2_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> holds <br/><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_10_1_1_1_2_1_2"/><i><font color="Green" title="E62">A63</font></i>: 
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>
 ;<br/>

<b>assume </b><a NAME="E2:61_10_1_1_1_2_1_2"/>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font>
 ;<br/>

<a NAME="E3:61_10_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:61_10_1_1_1_2_1_2"/>
<font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1_1_2_1_2"><i><font color="Green" title="E62">A63</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:61_10_1_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>]
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1"><i><font color="Green" title="E54">A55</font></i></a>, <a class="txt" href="neckla_3.html#E15:61_10_1_1_1"><i><font color="Green" title="E58">A59</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_1_1_2_1"><i><font color="Green" title="E61">A62</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:61_10_1_1_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:61_10_1_1_1"/>
 ex <font color="Olive" title="b1">n0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n0</font>] &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] holds <br/><font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n0</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="neckla_3.html#E17:61_10_1_1_1"><i><font color="Green" title="E60">A61</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c24">n0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E20:61_10_1_1_1"/><i><font color="Green" title="E61">A64</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c24">n0</font>]
 <b>and </b><br/><a NAME="E21:61_10_1_1_1"/><i><font color="Green" title="E62">A65</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">n0</font>
 ;<br/>
<a NAME="E22:61_10_1_1_1"/>
<font color="Maroon" title="c24">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_1_3"/>
not <font color="Maroon" title="c24">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:61_10_1_1_1_3"/><b>then </b>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_1_1"><i><font color="Green" title="E61">A64</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:61_10_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c25">k0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E24:61_10_1_1_1"/><i><font color="Green" title="E63">A66</font></i>: 
<font color="Maroon" title="c24">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<a NAME="E25:61_10_1_1_1"/><i><font color="Green" title="E64">A67</font></i>: 
<font color="Maroon" title="c24">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_1_4"/>
not <font color="Maroon" title="c24">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:61_10_1_1_1_4"/><b>then </b>
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font> &amp; <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E21:61"><i><font color="Green" title="E49">A31</font></i></a>, <a class="txt" href="neckla_3.html#E15:61_10_1_1_1"><i><font color="Green" title="E58">A59</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_10_1_1_1"><i><font color="Green" title="E61">A64</font></i></a></i>;<br/>
<a NAME="E3:61_10_1_1_1_4"/><b>then </b>
not <font color="Maroon" title="c13">X1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">Y1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<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="E4:61_10_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E14:61"><i><font color="Green" title="E43">A11</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="E26:61_10_1_1_1"/><i><font color="Green" title="E65">A68</font></i>: 
<font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_1_1_5"/><i><font color="Green" title="E66">A69</font></i>: 
<font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_1_5_1"/>
not <font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/>

<a NAME="E2:61_10_1_1_1_5_1"/><b>then </b>
<font color="Maroon" title="c25">k0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/>
<b>hence </b><a NAME="E3:61_10_1_1_1_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_1_1"><i><font color="Green" title="E63">A66</font></i></a>, <a class="txt" href="neckla_3.html#E25:61_10_1_1_1"><i><font color="Green" title="E64">A67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_10_1_1_1_5"/>
<font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_5_2"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_1_1_5_2"/><i><font color="Green" title="E67">A70</font></i>: 
<font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E2:61_10_1_1_1_5_2"/>
<font color="Maroon" title="c24">n0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_1_1"><i><font color="Green" title="E61">A64</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:61_10_1_1_1_5_2"/><b>then </b>
<font color="Maroon" title="c24">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:61_10_1_1_1_5_2"/>
<font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_1_1"><i><font color="Green" title="E63">A66</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_1_1_5_2"><i><font color="Green" title="E67">A70</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:61_10_1_1_1_5"/>
<font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1_1_5"><i><font color="Green" title="E66">A69</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:61_10_1_1_1"/>
<font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n0</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_1_1"><i><font color="Green" title="E63">A66</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E28:61_10_1_1_1"/><b>then </b><i><font color="Green" title="E66">A71</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c25">k0</font>]
 
<b>by </b><i><a class="txt" href="neckla_3.html#E21:61_10_1_1_1"><i><font color="Green" title="E62">A65</font></i></a></i>;<br/>
<a NAME="E29:61_10_1_1_1"/><i><font color="Green" title="E67">A72</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c25">k0</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> holds <br/><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_1_6"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c25">k0</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> &amp; not <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c26">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:61_10_1_1_1_6"/><i><font color="Green" title="E68">A73</font></i>: 
( <font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c25">k0</font> &amp; <font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> &amp; not <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font> )
 ;<br/>
<a NAME="E4:61_10_1_1_1_6"/>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">n0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_6_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_10_1_1_1_6_1_1"/>
( <font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n0</font> or <font color="Maroon" title="c24">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">k</font> or <font color="Maroon" title="c24">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:61_10_1_1_1_6_1_1_1"/>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n0</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_10_1_1_1_6_1_1_1"/>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">n0</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_1_1"><i><font color="Green" title="E63">A66</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_1_1_6"><i><font color="Green" title="E68">A73</font></i></a>, <a class="ref" href="nat_1.html#T25" title="NAT_1:th.25">NAT_1:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:61_10_1_1_1_6_1_1_2"/>
<font color="Maroon" title="c24">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_10_1_1_1_6_1_1_2"/>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">n0</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:61_10_1_1_1_6_1_1_3"/>
<font color="Maroon" title="c24">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_10_1_1_1_6_1_1_3"/>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">n0</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_1_1"><i><font color="Green" title="E61">A64</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_1_1_6"><i><font color="Green" title="E68">A73</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:61_10_1_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_1_1"><i><font color="Green" title="E61">A64</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_1_1_6"><i><font color="Green" title="E68">A73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:61_10_1_1_1"/><i><font color="Green" title="E68">A74</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E16:61_10_1_1_1"><i><font color="Green" title="E59">A60</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_10_1_1_1"><i><font color="Green" title="E61">A64</font></i></a>, <a class="txt" href="neckla_3.html#E24:61_10_1_1_1"><i><font color="Green" title="E63">A66</font></i></a>, <a class="txt" href="neckla_3.html#E26:61_10_1_1_1"><i><font color="Green" title="E65">A68</font></i></a></i>;<br/>
<a NAME="E31:61_10_1_1_1"/><i><font color="Green" title="E69">A75</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_1_1_7"/><i><font color="Green" title="E70">A76</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a></i>;<br/>
<a NAME="E2:61_10_1_1_1_7"/>
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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">G</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_1_1"><i><font color="Green" title="E63">A66</font></i></a>, <a class="txt" href="neckla_3.html#E30:61_10_1_1_1"><i><font color="Green" title="E68">A74</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E3:61_10_1_1_1_7"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_1_1"><i><font color="Green" title="E63">A66</font></i></a>, <a class="txt" href="neckla_3.html#E30:61_10_1_1_1"><i><font color="Green" title="E68">A74</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_1_1_7"><i><font color="Green" title="E70">A76</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:61_10_1_1_1"/>
<font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E30:61_10_1_1_1"><i><font color="Green" title="E68">A74</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E33:61_10_1_1_1"/><b>then </b>
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c11">R</font> or <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:61"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E34:61_10_1_1_1"/><b>then </b><i><font color="Green" title="E70">A77</font></i>: 
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font> or <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/>
<b>thus </b><a NAME="E35:61_10_1_1_1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Y1</font> ex <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">X1</font> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_1_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_10_1_1_1_8_1"/>
( ( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c3">R1</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> ) or ( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c4">R2</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> ) or ( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> ) )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_1_1"><i><font color="Green" title="E63">A66</font></i></a>, <a class="txt" href="neckla_3.html#E30:61_10_1_1_1"><i><font color="Green" title="E68">A74</font></i></a>, <a class="txt" href="neckla_3.html#E34:61_10_1_1_1"><i><font color="Green" title="E70">A77</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1_1_8_1_1"><b>suppose </b></a><a NAME="E1:61_10_1_1_1_8_1_1"/><i><font color="Green" title="E71">A78</font></i>: 
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c3">R1</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c26">m</font> = <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">X1</font> <b>by </b><i><a class="txt" href="neckla_3.html#E17:61"><i><font color="Green" title="E46">A20</font></i></a>, <a class="txt" href="neckla_3.html#E26:61_10_1_1_1"><i><font color="Green" title="E65">A68</font></i></a>, <a class="txt" href="neckla_3.html#E28:61_10_1_1_1"><i><font color="Green" title="E66">A71</font></i></a>, <a class="txt" href="neckla_3.html#E29:61_10_1_1_1"><i><font color="Green" title="E67">A72</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">l</font> = <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Y1</font> <b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_1_1"><i><font color="Green" title="E61">A64</font></i></a></i>;<br/><a NAME="E4:61_10_1_1_1_8_1_1"/><i><font color="Green" title="E72">A79</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">l</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E16:61_10_1_1_1"><i><font color="Green" title="E59">A60</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_10_1_1_1"><i><font color="Green" title="E61">A64</font></i></a>, <a class="txt" href="neckla_3.html#E24:61_10_1_1_1"><i><font color="Green" title="E63">A66</font></i></a>, <a class="txt" href="neckla_3.html#E26:61_10_1_1_1"><i><font color="Green" title="E65">A68</font></i></a></i>;<br/><a NAME="E5:61_10_1_1_1_8_1_1"/><i><font color="Green" title="E73">A80</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a></i>;<br/><a NAME="E6:61_10_1_1_1_8_1_1"/>
<font color="Maroon" title="c27">l</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="c3">R1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E17:61"><i><font color="Green" title="E46">A20</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_10_1_1_1"><i><font color="Green" title="E61">A64</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:61_10_1_1_1_8_1_1"/><b>then </b>
<font color="Maroon" title="c27">l</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</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/><a NAME="E8:61_10_1_1_1_8_1_1"/><b>then </b><i><font color="Green" title="E74">A81</font></i>: 
<font color="Maroon" title="c27">l</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="c11">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E9:61_10_1_1_1_8_1_1"/>
<font color="Maroon" title="c26">m</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1_1_8_1_1"><i><font color="Green" title="E71">A78</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E10:61_10_1_1_1_8_1_1"/><b>then </b><i><font color="Green" title="E75">A82</font></i>: 
<font color="Maroon" title="c26">m</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="c11">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E11:61_10_1_1_1_8_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:61"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E12:61_10_1_1_1_8_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">l</font>,<font color="Maroon" title="c26">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:61_10_1_1_1_8_1_1"><i><font color="Green" title="E72">A79</font></i></a>, <a class="txt" href="neckla_3.html#E5:61_10_1_1_1_8_1_1"><i><font color="Green" title="E73">A80</font></i></a>, <a class="txt" href="neckla_3.html#E8:61_10_1_1_1_8_1_1"><i><font color="Green" title="E74">A81</font></i></a>, <a class="txt" href="neckla_3.html#E10:61_10_1_1_1_8_1_1"><i><font color="Green" title="E75">A82</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/><b>hence </b><a NAME="E13:61_10_1_1_1_8_1_1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Y1</font> ex <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">X1</font> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1_1_8_1_2"><b>suppose </b></a><a NAME="E1:61_10_1_1_1_8_1_2"/>
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c4">R2</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c26">m</font> = <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font> ;<br/><b>reconsider </b><font color="Maroon" title="c27">l</font> = <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font> <b>by </b><i><a class="txt" href="neckla_3.html#E17:61"><i><font color="Green" title="E46">A20</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_10_1_1_1"><i><font color="Green" title="E61">A64</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:61_10_1_1_1_8_1_2"/>
<font color="Maroon" title="c27">l</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</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/><a NAME="E5:61_10_1_1_1_8_1_2"/><b>then </b><i><font color="Green" title="E71">A83</font></i>: 
<font color="Maroon" title="c27">l</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="c11">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E6:61_10_1_1_1_8_1_2"/>
<font color="Maroon" title="c26">m</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</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/><a NAME="E7:61_10_1_1_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c26">m</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="c11">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E8:61_10_1_1_1_8_1_2"/><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="c27">l</font>,<font color="Maroon" title="c26">m</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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#E5:61_10_1_1_1_8_1_2"><i><font color="Green" title="E71">A83</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:61_10_1_1_1_8_1_2"/><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="c27">l</font>,<font color="Maroon" title="c26">m</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E31:61_10_1_1_1"><i><font color="Green" title="E69">A75</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="E10:61_10_1_1_1_8_1_2"/><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="c27">l</font>,<font color="Maroon" title="c26">m</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c11">R</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>;<br/><b>hence </b><a NAME="E11:61_10_1_1_1_8_1_2"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Y1</font> ex <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">X1</font> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_3.html#T36" target="_self" title="NECKLA_3:th.36">Th36</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1_1_8_1_3"><b>suppose </b></a><a NAME="E1:61_10_1_1_1_8_1_3"/><i><font color="Green" title="E71">A84</font></i>: 
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c26">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_10_1_1_1_8_1_3"/><i><font color="Green" title="E72">A85</font></i>: 
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">y1</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c26">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_1_1"><i><font color="Green" title="E61">A64</font></i></a></i>;<br/><b>thus </b><a NAME="E4:61_10_1_1_1_8_1_3"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Y1</font> ex <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">X1</font> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E31:61_10_1_1_1"><i><font color="Green" title="E69">A75</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_1_1_8_1_3"><i><font color="Green" title="E71">A84</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_1_1_8_1_3"><i><font color="Green" title="E72">A85</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c19">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">Y1</font>, <font color="Maroon" title="c20">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">X1</font><b> such that </b><br/><a NAME="E4:61_10_1_1"/><i><font color="Green" title="E55">A86</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 ;<br/><b>consider </b><font color="Maroon" title="c21">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">X2</font>;<br/><b>set </b><font color="Maroon" title="c22">Z</font> = <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c21">w</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>;<br/><a NAME="E6:61_10_1_1"/>
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c21">w</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> <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">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">q</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="E1:61_10_1_1_2"/><i><font color="Green" title="E56">A87</font></i>: 
<font color="Maroon" title="c23">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c21">w</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 ;<br/>

<a NAME="E2:61_10_1_1_2"/><i><font color="Green" title="E57">A88</font></i>: 
( <font color="Maroon" title="c23">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font> or <font color="Maroon" title="c23">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">v</font> or <font color="Maroon" title="c23">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c23">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">w</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1_2"><i><font color="Green" title="E56">A87</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/>
<a NAME="E3:61_10_1_1_2"/>
<font color="Maroon" title="c19">u</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="c3">R1</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E17:61"><i><font color="Green" title="E46">A20</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_1"><i><font color="Green" title="E54">A55</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:61_10_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">u</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</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/>
<a NAME="E5:61_10_1_1_2"/><b>then </b><i><font color="Green" title="E58">A89</font></i>: 
<font color="Maroon" title="c19">u</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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E6:61_10_1_1_2"/><i><font color="Green" title="E59">A90</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:61"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:61_10_1_1_2"/>
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E22:61"><i><font color="Green" title="E50">A32</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E9:61_10_1_1_2"/><i><font color="Green" title="E60">A91</font></i>: 
( <font color="Maroon" title="c24">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">v</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/>
<a NAME="E10:61_10_1_1_2"/>
<font color="Maroon" title="c20">v</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_1_2"><i><font color="Green" title="E60">A91</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:61_10_1_1_2"/><b>then </b><i><font color="Green" title="E61">A92</font></i>: 
<font color="Maroon" title="c20">v</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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E12:61_10_1_1_2"/>
<font color="Maroon" title="c21">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E25:61"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c25">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E14:61_10_1_1_2"/><i><font color="Green" title="E62">A93</font></i>: 
( <font color="Maroon" title="c25">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">w</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/>
<a NAME="E15:61_10_1_1_2"/>
<font color="Maroon" title="c21">w</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E14:61_10_1_1_2"><i><font color="Green" title="E62">A93</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:61_10_1_1_2"/><b>then </b>
<font color="Maroon" title="c21">w</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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <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="E17:61_10_1_1_2"/>
<font color="Maroon" title="c23">q</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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:61_10_1_1_2"><i><font color="Green" title="E57">A88</font></i></a>, <a class="txt" href="neckla_3.html#E5:61_10_1_1_2"><i><font color="Green" title="E58">A89</font></i></a>, <a class="txt" href="neckla_3.html#E6:61_10_1_1_2"><i><font color="Green" title="E59">A90</font></i></a>, <a class="txt" href="neckla_3.html#E11:61_10_1_1_2"><i><font color="Green" title="E61">A92</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c23">Z</font> = <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c21">w</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E8:61_10_1_1"/><i><font color="Green" title="E56">A94</font></i>: 
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1"><i><font color="Green" title="E54">A55</font></i></a></i>;<br/><a NAME="E9:61_10_1_1"/><i><font color="Green" title="E57">A95</font></i>: 
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E22:61"><i><font color="Green" title="E50">A32</font></i></a></i>;<br/><a NAME="E10:61_10_1_1"/><i><font color="Green" title="E58">A96</font></i>: 
<font color="Maroon" title="c21">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E25:61"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">u</font> = <font color="Maroon" title="c19">u</font>, <font color="Maroon" title="c25">v</font> = <font color="Maroon" title="c20">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="neckla_3.html#E4:61_10_1_1"><i><font color="Green" title="E55">A86</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:61_10_1_1"/>
<font color="Maroon" title="c21">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E25:61"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E14:61_10_1_1"/><i><font color="Green" title="E59">A97</font></i>: 
( <font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">w</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c26">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c27">w</font> = <font color="Maroon" title="c21">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="neckla_3.html#E14:61_10_1_1"><i><font color="Green" title="E59">A97</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E16:61_10_1_1"/>
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font> &amp; <font color="Maroon" title="c25">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_1_3"/><i><font color="Green" title="E60">A98</font></i>: 
<font color="Maroon" title="c24">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_3_1"/><i><font color="Green" title="E61">A99</font></i>: 
not <font color="Maroon" title="c24">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_10_1_1_3_1"/><i><font color="Green" title="E62">A100</font></i>: 
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_1"><i><font color="Green" title="E56">A94</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E5:61_10_1_1_3_1"/><i><font color="Green" title="E63">A101</font></i>: 
( <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">y2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_1"><i><font color="Green" title="E57">A95</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:61_10_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1_3_1"><i><font color="Green" title="E61">A99</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_1_3_1"><i><font color="Green" title="E62">A100</font></i></a>, <a class="txt" href="neckla_3.html#E5:61_10_1_1_3_1"><i><font color="Green" title="E63">A101</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_10_1_1_3"/><i><font color="Green" title="E61">A102</font></i>: 
<font color="Maroon" title="c25">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_3_2"/><i><font color="Green" title="E62">A103</font></i>: 
not <font color="Maroon" title="c25">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_10_1_1_3_2"/><i><font color="Green" title="E63">A104</font></i>: 
( <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_1"><i><font color="Green" title="E57">A95</font></i></a></i>;<br/>
<a NAME="E4:61_10_1_1_3_2"/>
<font color="Maroon" title="c2">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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1_3_2"><i><font color="Green" title="E62">A103</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_1_3_2"><i><font color="Green" title="E63">A104</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:61_10_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c2">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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E6:61_10_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c2">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">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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/>
<a NAME="E7:61_10_1_1_3_2"/><b>then </b>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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>hence </b><a NAME="E8:61_10_1_1_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:61_10_1_1_3"/><i><font color="Green" title="E62">A105</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_3_3"/><i><font color="Green" title="E63">A106</font></i>: 
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_10_1_1_3_3"/><i><font color="Green" title="E64">A107</font></i>: 
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_1"><i><font color="Green" title="E56">A94</font></i></a></i>;<br/>
<a NAME="E4:61_10_1_1_3_3"/>
<font color="Maroon" title="c2">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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1_3_3"><i><font color="Green" title="E63">A106</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_1_3_3"><i><font color="Green" title="E64">A107</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:61_10_1_1_3_3"/><b>then </b>
<font color="Maroon" title="c2">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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E6:61_10_1_1_3_3"/><b>then </b>
<font color="Maroon" title="c2">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">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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/>
<a NAME="E7:61_10_1_1_3_3"/><b>then </b>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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>hence </b><a NAME="E8:61_10_1_1_3_3"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:61_10_1_1_3"/><i><font color="Green" title="E63">A108</font></i>: 
<font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_3_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_3_4"/><i><font color="Green" title="E64">A109</font></i>: 
not <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_10_1_1_3_4"/><i><font color="Green" title="E65">A110</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_1"><i><font color="Green" title="E58">A96</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E5:61_10_1_1_3_4"/><i><font color="Green" title="E66">A111</font></i>: 
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">y2</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_1"><i><font color="Green" title="E56">A94</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:61_10_1_1_3_4"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1_3_4"><i><font color="Green" title="E64">A109</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_1_3_4"><i><font color="Green" title="E65">A110</font></i></a>, <a class="txt" href="neckla_3.html#E5:61_10_1_1_3_4"><i><font color="Green" title="E66">A111</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:61_10_1_1_3"/><i><font color="Green" title="E64">A112</font></i>: 
<font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_3_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_3_5"/><i><font color="Green" title="E65">A113</font></i>: 
not <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_10_1_1_3_5"/><i><font color="Green" title="E66">A114</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_1"><i><font color="Green" title="E58">A96</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E5:61_10_1_1_3_5"/><i><font color="Green" title="E67">A115</font></i>: 
( <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">y2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_1"><i><font color="Green" title="E57">A95</font></i></a></i>;<br/>
<a NAME="E6:61_10_1_1_3_5"/>
<font color="Maroon" title="c28">y1</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="c3">R1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1_3_5"><i><font color="Green" title="E65">A113</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_1_3_5"><i><font color="Green" title="E66">A114</font></i></a>, <a class="txt" href="neckla_3.html#E5:61_10_1_1_3_5"><i><font color="Green" title="E67">A115</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="E7:61_10_1_1_3_5"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</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="E6:61_10_1_1_3"/>
<font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_3_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_3_6"/><i><font color="Green" title="E65">A116</font></i>: 
not <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_10_1_1_3_6"/><i><font color="Green" title="E66">A117</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_1"><i><font color="Green" title="E58">A96</font></i></a></i>;<br/>
<a NAME="E4:61_10_1_1_3_6"/>
<font color="Maroon" title="c2">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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1_3_6"><i><font color="Green" title="E65">A116</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_1_3_6"><i><font color="Green" title="E66">A117</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:61_10_1_1_3_6"/><b>then </b>
<font color="Maroon" title="c2">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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E6:61_10_1_1_3_6"/><b>then </b>
<font color="Maroon" title="c2">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">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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/>
<a NAME="E7:61_10_1_1_3_6"/><b>then </b>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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>hence </b><a NAME="E8:61_10_1_1_3_6"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:61_10_1_1_3"/>
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font> &amp; <font color="Maroon" title="c25">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_1_3"><i><font color="Green" title="E60">A98</font></i></a>, <a class="txt" href="neckla_3.html#E2:61_10_1_1_3"><i><font color="Green" title="E61">A102</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_1_3"><i><font color="Green" title="E62">A105</font></i></a>, <a class="txt" href="neckla_3.html#E4:61_10_1_1_3"><i><font color="Green" title="E63">A108</font></i></a>, <a class="txt" href="neckla_3.html#E5:61_10_1_1_3"><i><font color="Green" title="E64">A112</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:61_10_1_1"/><b>then </b><i><font color="Green" title="E60">A118</font></i>: 
<font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c25">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c27">w</font> <a href="incproj.html#R2" title="INCPROJ:pred.2">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="incproj.html#D6" title="INCPROJ:def.6">INCPROJ:def 6</a></i>;<br/><a NAME="E18:61_10_1_1"/><i><font color="Green" title="E61">A119</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">v</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_4"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E2:61_10_1_1_4"/><i><font color="Green" title="E62">A120</font></i>: 
( <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_1"><i><font color="Green" title="E57">A95</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:61_10_1_1_4"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">v</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:61_10_1_1_4"><i><font color="Green" title="E62">A120</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:61_10_1_1"/><i><font color="Green" title="E62">A121</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E2:61_10_1_1_5"/><i><font color="Green" title="E63">A122</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_1"><i><font color="Green" title="E58">A96</font></i></a></i>;<br/>
<a NAME="E3:61_10_1_1_5"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:61_10_1_1_5"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:61_10_1_1_5"><i><font color="Green" title="E63">A122</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:61_10_1_1"/><i><font color="Green" title="E63">A123</font></i>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_6"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E2:61_10_1_1_6"/><i><font color="Green" title="E64">A124</font></i>: 
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_1"><i><font color="Green" title="E56">A94</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:61_10_1_1_6"/>
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:61_10_1_1_6"><i><font color="Green" title="E64">A124</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:61_10_1_1"/><i><font color="Green" title="E64">A125</font></i>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_7"/><i><font color="Green" title="E65">A126</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">u</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_10_1_1_7"/><i><font color="Green" title="E66">A127</font></i>: 
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_1"><i><font color="Green" title="E56">A94</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E5:61_10_1_1_7"/><i><font color="Green" title="E67">A128</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">y2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_1"><i><font color="Green" title="E58">A96</font></i></a></i>;<br/>
<a NAME="E6:61_10_1_1_7"/>
<font color="Maroon" title="c24">u</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:61_10_1_1_7"><i><font color="Green" title="E66">A127</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">u</font> = <font color="Maroon" title="c24">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E8:61_10_1_1_7"/>
<font color="Maroon" title="c27">w</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:61_10_1_1_7"><i><font color="Green" title="E67">A128</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c31">w</font> = <font color="Maroon" title="c27">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E10:61_10_1_1_7"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:61_10_1_1_7"/><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="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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#E1:61_10_1_1_7"><i><font color="Green" title="E65">A126</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="E12:61_10_1_1_7"/><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="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>
 
;<br/>
<a NAME="E13:61_10_1_1_7"/><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="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c11">R</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>;<br/>
<a NAME="E14:61_10_1_1_7"/><b>then </b><i><font color="Green" title="E68">A129</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font> <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 <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_10_1_1_7_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">R2</font> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E14:61_10_1_1_7"><i><font color="Green" title="E68">A129</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="61_10_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:61_10_1_1_7_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font>
 ;<br/><div class="add"><a NAME="E2:61_10_1_1_7_1_1"/><b>then </b>
( <font color="Maroon" title="c30">u</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="c3">R1</font> &amp; <font color="Maroon" title="c31">w</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="c3">R1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:61_10_1_1_7_1_1"/><b>then </b>
<font color="Maroon" title="c31">w</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="c3">R1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:61_10_1_1_7"><i><font color="Green" title="E67">A128</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="E4:61_10_1_1_7_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</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="61_10_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:61_10_1_1_7_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">R2</font>
 ;<br/><div class="add"><a NAME="E2:61_10_1_1_7_1_2"/><b>then </b>
( <font color="Maroon" title="c30">u</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="c4">R2</font> &amp; <font color="Maroon" title="c31">w</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="c4">R2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:61_10_1_1_7_1_2"/><b>then </b>
<font color="Maroon" title="c30">u</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="c3">R1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_10_1_1_7"><i><font color="Green" title="E66">A127</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="E4:61_10_1_1_7_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</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><b>end;</b></div>

</div><b>end;</b></div><a NAME="E22:61_10_1_1"/>
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">v</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_8"/><i><font color="Green" title="E65">A130</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">v</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_10_1_1_8"/><i><font color="Green" title="E66">A131</font></i>: 
( <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_1"><i><font color="Green" title="E57">A95</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E5:61_10_1_1_8"/><i><font color="Green" title="E67">A132</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">y2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_1"><i><font color="Green" title="E58">A96</font></i></a></i>;<br/>
<a NAME="E6:61_10_1_1_8"/>
<font color="Maroon" title="c25">v</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:61_10_1_1_8"><i><font color="Green" title="E66">A131</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">v</font> = <font color="Maroon" title="c25">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E8:61_10_1_1_8"/>
<font color="Maroon" title="c27">w</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:61_10_1_1_8"><i><font color="Green" title="E67">A132</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c31">w</font> = <font color="Maroon" title="c27">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E10:61_10_1_1_8"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:61_10_1_1_8"/><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="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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#E1:61_10_1_1_8"><i><font color="Green" title="E65">A130</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="E12:61_10_1_1_8"/><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="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>
 
;<br/>
<a NAME="E13:61_10_1_1_8"/><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="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c11">R</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>;<br/>
<a NAME="E14:61_10_1_1_8"/><b>then </b><i><font color="Green" title="E68">A133</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font> <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 <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_10_1_1_8_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">R2</font> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E14:61_10_1_1_8"><i><font color="Green" title="E68">A133</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="61_10_1_1_8_1_1"><b>suppose </b></a><a NAME="E1:61_10_1_1_8_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font>
 ;<br/><div class="add"><a NAME="E2:61_10_1_1_8_1_1"/><b>then </b>
( <font color="Maroon" title="c30">v</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="c3">R1</font> &amp; <font color="Maroon" title="c31">w</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="c3">R1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:61_10_1_1_8_1_1"/><b>then </b>
<font color="Maroon" title="c31">w</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="c3">R1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:61_10_1_1_8"><i><font color="Green" title="E67">A132</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="E4:61_10_1_1_8_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</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="61_10_1_1_8_1_2"><b>suppose </b></a><a NAME="E1:61_10_1_1_8_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">R2</font>
 ;<br/><div class="add"><a NAME="E2:61_10_1_1_8_1_2"/><b>then </b>
( <font color="Maroon" title="c30">v</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="c4">R2</font> &amp; <font color="Maroon" title="c31">w</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="c4">R2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:61_10_1_1_8_1_2"/><b>then </b>
<font color="Maroon" title="c30">v</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="c3">R1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_10_1_1_8"><i><font color="Green" title="E66">A131</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="E4:61_10_1_1_8_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</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><b>end;</b></div>

</div><b>end;</b></div><a NAME="E23:61_10_1_1"/><b>then </b><i><font color="Green" title="E65">A134</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c23">Z</font> <a href="necklace.html#R1" title="NECKLACE:pred.1">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:61_10_1_1"><i><font color="Green" title="E55">A86</font></i></a>, <a class="txt" href="neckla_3.html#E17:61_10_1_1"><i><font color="Green" title="E60">A118</font></i></a>, <a class="txt" href="neckla_3.html#E18:61_10_1_1"><i><font color="Green" title="E61">A119</font></i></a>, <a class="txt" href="neckla_3.html#E19:61_10_1_1"><i><font color="Green" title="E62">A121</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_10_1_1"><i><font color="Green" title="E63">A123</font></i></a>, <a class="txt" href="neckla_3.html#E21:61_10_1_1"><i><font color="Green" title="E64">A125</font></i></a>, <a class="ref" href="neckla_3.html#T39" target="_self" title="NECKLA_3:th.39">Th39</a></i>;<br/><a NAME="E24:61_10_1_1"/>
not  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c23">Z</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_9"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_1_9"/>
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="c23">Z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Z</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="E2:61_10_1_1_9"/><b>then </b>
<font color="Maroon" title="c24">u</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c23">Z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:61_10_1_1_9"/>
not  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c23">Z</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c28">H</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c23">Z</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E26:61_10_1_1"/>
<font color="Maroon" title="c1">G</font> <a href="necklace.html#R2" title="NECKLACE:pred.2">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_1_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_1_10"/>
not <font color="Maroon" title="c1">G</font> <a href="necklace.html#R2" title="NECKLACE:pred.2">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4
 ;<br/>

<a NAME="E2:61_10_1_1_10"/><b>then </b>
<font color="Maroon" title="c1">G</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 
<b>by </b><i><a class="ref" href="neckla_2.html#D1" title="NECKLA_2:def.1">NECKLA_2:def 1</a></i>;<br/>
<a NAME="E3:61_10_1_1_10"/><b>then </b>
<font color="Maroon" title="c28">H</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 
<b>by </b><i><a class="ref" href="neckla_3.html#T23" target="_self" title="NECKLA_3:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E4:61_10_1_1_10"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E23:61_10_1_1"><i><font color="Green" title="E65">A134</font></i></a>, <a class="ref" href="neckla_2.html#D1" title="NECKLA_2:def.1">NECKLA_2:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E27:61_10_1_1"/>
<font color="Maroon" title="c1">G</font> <a href="necklace.html#R2" title="NECKLACE:pred.2">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2"><b>suppose </b></a><a NAME="E1:61_10_1_2"/><i><font color="Green" title="E54">A135</font></i>: 
not <font color="Maroon" title="c16">Y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:61_10_1_2"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">Y2</font> ex <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">X2</font> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c19">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">Y2</font>;<br/>
<a NAME="E2:61_10_1_2_1"/>
<font color="Maroon" title="c18">b</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</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/>
<a NAME="E3:61_10_1_2_1"/><b>then </b><i><font color="Green" title="E55">A136</font></i>: 
<font color="Maroon" title="c18">b</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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E4:61_10_1_2_1"/><i><font color="Green" title="E56">A137</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:61"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">b</font> = <font color="Maroon" title="c18">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_10_1_2_1"><i><font color="Green" title="E55">A136</font></i></a></i>;<br/>
<a NAME="E6:61_10_1_2_1"/>
<font color="Maroon" title="c19">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2"><i><font color="Green" title="E54">A135</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E8:61_10_1_2_1"/><i><font color="Green" title="E57">A138</font></i>: 
( <font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">c</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c21">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/>
<a NAME="E9:61_10_1_2_1"/>
<font color="Maroon" title="c19">c</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_2_1"><i><font color="Green" title="E57">A138</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E10:61_10_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">c</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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">c</font> = <font color="Maroon" title="c19">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="neckla_3.html#E4:61_10_1_2_1"><i><font color="Green" title="E56">A137</font></i></a></i>;<br/>
<a NAME="E12:61_10_1_2_1"/>
<font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_1_1"/>
not <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c22">c</font>
 ;<br/>

<a NAME="E2:61_10_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c22">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font> &amp; <font color="Maroon" title="c22">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E24:61"><i><font color="Green" title="E51">A33</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_2"><i><font color="Green" title="E54">A135</font></i></a></i>;<br/>
<a NAME="E3:61_10_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c22">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">Y2</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="E4:61_10_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E15:61"><i><font color="Green" title="E44">A15</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="E13:61_10_1_2_1"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c20">b</font>,<font color="Maroon" title="c22">c</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:61"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="neckla_3.html#D2" target="_self" title="NECKLA_3:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E15:61_10_1_2_1"/><i><font color="Green" title="E58">A139</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">c</font> )
 <b>and </b><br/><a NAME="E16:61_10_1_2_1"/><i><font color="Green" title="E59">A140</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T12" title="REWRITE1:th.12">REWRITE1:12</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> $1 &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> holds <br/><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font> ) );<br/>
<a NAME="E17:61_10_1_2_1"/><i><font color="Green" title="E60">A141</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_2_1_2_1"/><i><font color="Green" title="E61">A142</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_2_1_2_1_1"/>
<font color="Maroon" title="c23">p</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="txt" href="neckla_3.html#E15:61_10_1_2_1"><i><font color="Green" title="E58">A139</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E2:61_10_1_2_1_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_10_1_2_1_2_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> holds <br/><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:61_10_1_2_1_2_1_2"/><i><font color="Green" title="E62">A143</font></i>: 
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>
 ;<br/>

<b>assume </b><a NAME="E2:61_10_1_2_1_2_1_2"/>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font>
 ;<br/>

<a NAME="E3:61_10_1_2_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:61_10_1_2_1_2_1_2"/>
<font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2_1_2_1_2"><i><font color="Green" title="E62">A143</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:61_10_1_2_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>]
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2"><i><font color="Green" title="E54">A135</font></i></a>, <a class="txt" href="neckla_3.html#E15:61_10_1_2_1"><i><font color="Green" title="E58">A139</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_2_1_2_1"><i><font color="Green" title="E61">A142</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:61_10_1_2_1_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:61_10_1_2_1"/>
 ex <font color="Olive" title="b1">n0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n0</font>] &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] holds <br/><font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n0</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="neckla_3.html#E17:61_10_1_2_1"><i><font color="Green" title="E60">A141</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c24">n0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E20:61_10_1_2_1"/><i><font color="Green" title="E61">A144</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c24">n0</font>]
 <b>and </b><br/><a NAME="E21:61_10_1_2_1"/><i><font color="Green" title="E62">A145</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">n0</font>
 ;<br/>
<a NAME="E22:61_10_1_2_1"/>
<font color="Maroon" title="c24">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_1_3"/>
not <font color="Maroon" title="c24">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:61_10_1_2_1_3"/><b>then </b>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_2_1"><i><font color="Green" title="E61">A144</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:61_10_1_2_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c25">k0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E24:61_10_1_2_1"/><i><font color="Green" title="E63">A146</font></i>: 
<font color="Maroon" title="c24">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<a NAME="E25:61_10_1_2_1"/><i><font color="Green" title="E64">A147</font></i>: 
<font color="Maroon" title="c24">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_1_4"/>
not <font color="Maroon" title="c24">n0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:61_10_1_2_1_4"/><b>then </b>
( <font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E24:61"><i><font color="Green" title="E51">A33</font></i></a>, <a class="txt" href="neckla_3.html#E15:61_10_1_2_1"><i><font color="Green" title="E58">A139</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_10_1_2_1"><i><font color="Green" title="E61">A144</font></i></a></i>;<br/>
<a NAME="E3:61_10_1_2_1_4"/><b>then </b>
not <font color="Maroon" title="c15">X2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">Y2</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<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="E4:61_10_1_2_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E15:61"><i><font color="Green" title="E44">A15</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="E26:61_10_1_2_1"/><i><font color="Green" title="E65">A148</font></i>: 
<font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_2_1_5"/><i><font color="Green" title="E66">A149</font></i>: 
<font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_1_5_1"/>
not <font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/>

<a NAME="E2:61_10_1_2_1_5_1"/><b>then </b>
<font color="Maroon" title="c25">k0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/>
<b>hence </b><a NAME="E3:61_10_1_2_1_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_2_1"><i><font color="Green" title="E63">A146</font></i></a>, <a class="txt" href="neckla_3.html#E25:61_10_1_2_1"><i><font color="Green" title="E64">A147</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_10_1_2_1_5"/>
<font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_5_2"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_2_1_5_2"/><i><font color="Green" title="E67">A150</font></i>: 
<font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E2:61_10_1_2_1_5_2"/>
<font color="Maroon" title="c24">n0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_2_1"><i><font color="Green" title="E61">A144</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:61_10_1_2_1_5_2"/><b>then </b>
<font color="Maroon" title="c24">n0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:61_10_1_2_1_5_2"/>
<font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_2_1"><i><font color="Green" title="E63">A146</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_2_1_5_2"><i><font color="Green" title="E67">A150</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:61_10_1_2_1_5"/><b>then </b>
<font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2_1_5"><i><font color="Green" title="E66">A149</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:61_10_1_2_1_5"/>
<font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:61_10_1_2_1"/>
<font color="Maroon" title="c25">k0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n0</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_2_1"><i><font color="Green" title="E63">A146</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E28:61_10_1_2_1"/><b>then </b><i><font color="Green" title="E66">A151</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c25">k0</font>]
 
<b>by </b><i><a class="txt" href="neckla_3.html#E21:61_10_1_2_1"><i><font color="Green" title="E62">A145</font></i></a></i>;<br/>
<a NAME="E29:61_10_1_2_1"/><i><font color="Green" title="E67">A152</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c25">k0</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> holds <br/><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_1_6"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c25">k0</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> &amp; not <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c26">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:61_10_1_2_1_6"/><i><font color="Green" title="E68">A153</font></i>: 
( <font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c25">k0</font> &amp; <font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">p</font> &amp; not <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font> )
 ;<br/>
<a NAME="E4:61_10_1_2_1_6"/>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">n0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_6_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_10_1_2_1_6_1_1"/>
( <font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n0</font> or <font color="Maroon" title="c24">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">k</font> or <font color="Maroon" title="c24">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:61_10_1_2_1_6_1_1_1"/>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n0</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_10_1_2_1_6_1_1_1"/>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">n0</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_2_1"><i><font color="Green" title="E63">A146</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_2_1_6"><i><font color="Green" title="E68">A153</font></i></a>, <a class="ref" href="nat_1.html#T25" title="NAT_1:th.25">NAT_1:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:61_10_1_2_1_6_1_1_2"/>
<font color="Maroon" title="c24">n0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_10_1_2_1_6_1_1_2"/>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">n0</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:61_10_1_2_1_6_1_1_3"/>
<font color="Maroon" title="c24">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_10_1_2_1_6_1_1_3"/>
<font color="Maroon" title="c26">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">n0</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_2_1"><i><font color="Green" title="E61">A144</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_2_1_6"><i><font color="Green" title="E68">A153</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:61_10_1_2_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_2_1"><i><font color="Green" title="E61">A144</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_2_1_6"><i><font color="Green" title="E68">A153</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:61_10_1_2_1"/><i><font color="Green" title="E68">A154</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">k0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E16:61_10_1_2_1"><i><font color="Green" title="E59">A140</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_10_1_2_1"><i><font color="Green" title="E61">A144</font></i></a>, <a class="txt" href="neckla_3.html#E24:61_10_1_2_1"><i><font color="Green" title="E63">A146</font></i></a>, <a class="txt" href="neckla_3.html#E26:61_10_1_2_1"><i><font color="Green" title="E65">A148</font></i></a></i>;<br/>
<a NAME="E31:61_10_1_2_1"/><i><font color="Green" title="E69">A155</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_2_1_7"/><i><font color="Green" title="E70">A156</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a></i>;<br/>
<a NAME="E2:61_10_1_2_1_7"/>
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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">G</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_2_1"><i><font color="Green" title="E63">A146</font></i></a>, <a class="txt" href="neckla_3.html#E30:61_10_1_2_1"><i><font color="Green" title="E68">A154</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E3:61_10_1_2_1_7"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_2_1"><i><font color="Green" title="E63">A146</font></i></a>, <a class="txt" href="neckla_3.html#E30:61_10_1_2_1"><i><font color="Green" title="E68">A154</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_2_1_7"><i><font color="Green" title="E70">A156</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:61_10_1_2_1"/>
<font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E30:61_10_1_2_1"><i><font color="Green" title="E68">A154</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E33:61_10_1_2_1"/><b>then </b>
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c11">R</font> or <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:61"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E34:61_10_1_2_1"/><b>then </b><i><font color="Green" title="E70">A157</font></i>: 
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font> or <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/>
<b>thus </b><a NAME="E35:61_10_1_2_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">Y2</font> ex <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">X2</font> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_1_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_10_1_2_1_8_1"/>
( ( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c4">R2</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> ) or ( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c3">R1</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> ) or ( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> ) )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E24:61_10_1_2_1"><i><font color="Green" title="E63">A146</font></i></a>, <a class="txt" href="neckla_3.html#E30:61_10_1_2_1"><i><font color="Green" title="E68">A154</font></i></a>, <a class="txt" href="neckla_3.html#E34:61_10_1_2_1"><i><font color="Green" title="E70">A157</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2_1_8_1_1"><b>suppose </b></a><a NAME="E1:61_10_1_2_1_8_1_1"/>
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c4">R2</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c26">m</font> = <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">X2</font> <b>by </b><i><a class="txt" href="neckla_3.html#E18:61"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="neckla_3.html#E26:61_10_1_2_1"><i><font color="Green" title="E65">A148</font></i></a>, <a class="txt" href="neckla_3.html#E28:61_10_1_2_1"><i><font color="Green" title="E66">A151</font></i></a>, <a class="txt" href="neckla_3.html#E29:61_10_1_2_1"><i><font color="Green" title="E67">A152</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">l</font> = <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">Y2</font> <b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_2_1"><i><font color="Green" title="E61">A144</font></i></a></i>;<br/><a NAME="E4:61_10_1_2_1_8_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">m</font>,<font color="Maroon" title="c27">l</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E16:61_10_1_2_1"><i><font color="Green" title="E59">A140</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_10_1_2_1"><i><font color="Green" title="E61">A144</font></i></a>, <a class="txt" href="neckla_3.html#E24:61_10_1_2_1"><i><font color="Green" title="E63">A146</font></i></a>, <a class="txt" href="neckla_3.html#E26:61_10_1_2_1"><i><font color="Green" title="E65">A148</font></i></a></i>;<br/><b>hence </b><a NAME="E5:61_10_1_2_1_8_1_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">Y2</font> ex <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">X2</font> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E31:61_10_1_2_1"><i><font color="Green" title="E69">A155</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2_1_8_1_2"><b>suppose </b></a><a NAME="E1:61_10_1_2_1_8_1_2"/>
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</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="c3">R1</font> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c26">m</font> = <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font> ;<br/><b>reconsider </b><font color="Maroon" title="c27">l</font> = <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font> <b>by </b><i><a class="txt" href="neckla_3.html#E18:61"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_10_1_2_1"><i><font color="Green" title="E61">A144</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:61_10_1_2_1_8_1_2"/>
<font color="Maroon" title="c27">l</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</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/><a NAME="E5:61_10_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E71">A158</font></i>: 
<font color="Maroon" title="c27">l</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="c11">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E6:61_10_1_2_1_8_1_2"/>
<font color="Maroon" title="c26">m</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</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/><a NAME="E7:61_10_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E72">A159</font></i>: 
<font color="Maroon" title="c26">m</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="c11">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E8:61_10_1_2_1_8_1_2"/><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="c27">l</font>,<font color="Maroon" title="c26">m</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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#E5:61_10_1_2_1_8_1_2"><i><font color="Green" title="E71">A158</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:61_10_1_2_1_8_1_2"/><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="c27">l</font>,<font color="Maroon" title="c26">m</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E31:61_10_1_2_1"><i><font color="Green" title="E69">A155</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="E10:61_10_1_2_1_8_1_2"/><b>then </b><i><font color="Green" title="E73">A160</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c27">l</font>,<font color="Maroon" title="c26">m</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c11">R</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>;<br/><a NAME="E11:61_10_1_2_1_8_1_2"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c11">R</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>
 <b>by </b><i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a></i>;<br/><a NAME="E12:61_10_1_2_1_8_1_2"/><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="c26">m</font>,<font color="Maroon" title="c27">l</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c11">R</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:61_10_1_2_1_8_1_2"><i><font color="Green" title="E71">A158</font></i></a>, <a class="txt" href="neckla_3.html#E7:61_10_1_2_1_8_1_2"><i><font color="Green" title="E72">A159</font></i></a>, <a class="txt" href="neckla_3.html#E10:61_10_1_2_1_8_1_2"><i><font color="Green" title="E73">A160</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/><b>hence </b><a NAME="E13:61_10_1_2_1_8_1_2"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">Y2</font> ex <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">X2</font> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_3.html#T36" target="_self" title="NECKLA_3:th.36">Th36</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2_1_8_1_3"><b>suppose </b></a><a NAME="E1:61_10_1_2_1_8_1_3"/><i><font color="Green" title="E71">A161</font></i>: 
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</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">G</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c26">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_10_1_2_1_8_1_3"/><i><font color="Green" title="E72">A162</font></i>: 
( <font color="Maroon" title="c23">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">y1</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c26">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E20:61_10_1_2_1"><i><font color="Green" title="E61">A144</font></i></a></i>;<br/><b>thus </b><a NAME="E4:61_10_1_2_1_8_1_3"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">Y2</font> ex <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">X2</font> st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E31:61_10_1_2_1"><i><font color="Green" title="E69">A155</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_2_1_8_1_3"><i><font color="Green" title="E71">A161</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_2_1_8_1_3"><i><font color="Green" title="E72">A162</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c19">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">Y2</font>, <font color="Maroon" title="c20">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">X2</font><b> such that </b><br/><a NAME="E4:61_10_1_2"/><i><font color="Green" title="E55">A163</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">G</font>
 ;<br/><b>consider </b><font color="Maroon" title="c21">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">X1</font>;<br/><b>set </b><font color="Maroon" title="c22">Z</font> = <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c21">w</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>;<br/><a NAME="E6:61_10_1_2"/>
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c21">w</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> <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">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">q</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="E1:61_10_1_2_2"/><i><font color="Green" title="E56">A164</font></i>: 
<font color="Maroon" title="c23">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c21">w</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 ;<br/>

<a NAME="E2:61_10_1_2_2"/><i><font color="Green" title="E57">A165</font></i>: 
( <font color="Maroon" title="c23">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</font> or <font color="Maroon" title="c23">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">v</font> or <font color="Maroon" title="c23">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c23">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">w</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2_2"><i><font color="Green" title="E56">A164</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/>
<a NAME="E3:61_10_1_2_2"/>
<font color="Maroon" title="c19">u</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="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E18:61"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="neckla_3.html#E1:61_10_1_2"><i><font color="Green" title="E54">A135</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:61_10_1_2_2"/><b>then </b>
<font color="Maroon" title="c19">u</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</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/>
<a NAME="E5:61_10_1_2_2"/><b>then </b><i><font color="Green" title="E58">A166</font></i>: 
<font color="Maroon" title="c19">u</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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E6:61_10_1_2_2"/><i><font color="Green" title="E59">A167</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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">G</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:61"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:61_10_1_2_2"/>
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E25:61"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E9:61_10_1_2_2"/><i><font color="Green" title="E60">A168</font></i>: 
( <font color="Maroon" title="c24">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">v</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/>
<a NAME="E10:61_10_1_2_2"/>
<font color="Maroon" title="c20">v</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_2_2"><i><font color="Green" title="E60">A168</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:61_10_1_2_2"/><b>then </b><i><font color="Green" title="E61">A169</font></i>: 
<font color="Maroon" title="c20">v</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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E12:61_10_1_2_2"/>
<font color="Maroon" title="c21">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E22:61"><i><font color="Green" title="E50">A32</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c25">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E14:61_10_1_2_2"/><i><font color="Green" title="E62">A170</font></i>: 
( <font color="Maroon" title="c25">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">w</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/>
<a NAME="E15:61_10_1_2_2"/>
<font color="Maroon" title="c21">w</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E14:61_10_1_2_2"><i><font color="Green" title="E62">A170</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E16:61_10_1_2_2"/><b>then </b>
<font color="Maroon" title="c21">w</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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <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="E17:61_10_1_2_2"/>
<font color="Maroon" title="c23">q</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">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:61_10_1_2_2"><i><font color="Green" title="E57">A165</font></i></a>, <a class="txt" href="neckla_3.html#E5:61_10_1_2_2"><i><font color="Green" title="E58">A166</font></i></a>, <a class="txt" href="neckla_3.html#E6:61_10_1_2_2"><i><font color="Green" title="E59">A167</font></i></a>, <a class="txt" href="neckla_3.html#E11:61_10_1_2_2"><i><font color="Green" title="E61">A169</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c23">Z</font> = <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c21">w</font></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E8:61_10_1_2"/><i><font color="Green" title="E56">A171</font></i>: 
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2"><i><font color="Green" title="E54">A135</font></i></a></i>;<br/><a NAME="E9:61_10_1_2"/><i><font color="Green" title="E57">A172</font></i>: 
<font color="Maroon" title="c20">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">X2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E25:61"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/><a NAME="E10:61_10_1_2"/><i><font color="Green" title="E58">A173</font></i>: 
<font color="Maroon" title="c21">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E22:61"><i><font color="Green" title="E50">A32</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">u</font> = <font color="Maroon" title="c19">u</font>, <font color="Maroon" title="c25">v</font> = <font color="Maroon" title="c20">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="neckla_3.html#E4:61_10_1_2"><i><font color="Green" title="E55">A163</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:61_10_1_2"/>
<font color="Maroon" title="c21">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E22:61"><i><font color="Green" title="E50">A32</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E14:61_10_1_2"/><i><font color="Green" title="E59">A174</font></i>: 
( <font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">w</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c26">y</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c27">w</font> = <font color="Maroon" title="c21">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="neckla_3.html#E14:61_10_1_2"><i><font color="Green" title="E59">A174</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E16:61_10_1_2"/>
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font> &amp; <font color="Maroon" title="c25">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_2_3"/><i><font color="Green" title="E60">A175</font></i>: 
<font color="Maroon" title="c24">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_3_1"/><i><font color="Green" title="E61">A176</font></i>: 
not <font color="Maroon" title="c24">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_10_1_2_3_1"/><i><font color="Green" title="E62">A177</font></i>: 
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_2"><i><font color="Green" title="E56">A171</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E5:61_10_1_2_3_1"/><i><font color="Green" title="E63">A178</font></i>: 
( <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">y2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_2"><i><font color="Green" title="E57">A172</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:61_10_1_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2_3_1"><i><font color="Green" title="E61">A176</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_2_3_1"><i><font color="Green" title="E62">A177</font></i></a>, <a class="txt" href="neckla_3.html#E5:61_10_1_2_3_1"><i><font color="Green" title="E63">A178</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61_10_1_2_3"/><i><font color="Green" title="E61">A179</font></i>: 
<font color="Maroon" title="c25">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_3_2"/><i><font color="Green" title="E62">A180</font></i>: 
not <font color="Maroon" title="c25">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_10_1_2_3_2"/><i><font color="Green" title="E63">A181</font></i>: 
( <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_2"><i><font color="Green" title="E57">A172</font></i></a></i>;<br/>
<a NAME="E4:61_10_1_2_3_2"/>
<font color="Maroon" title="c2">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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2_3_2"><i><font color="Green" title="E62">A180</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_2_3_2"><i><font color="Green" title="E63">A181</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:61_10_1_2_3_2"/><b>then </b>
<font color="Maroon" title="c2">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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E6:61_10_1_2_3_2"/><b>then </b>
<font color="Maroon" title="c2">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">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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/>
<a NAME="E7:61_10_1_2_3_2"/><b>then </b>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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>hence </b><a NAME="E8:61_10_1_2_3_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:61_10_1_2_3"/><i><font color="Green" title="E62">A182</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_3_3"/><i><font color="Green" title="E63">A183</font></i>: 
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_10_1_2_3_3"/><i><font color="Green" title="E64">A184</font></i>: 
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_2"><i><font color="Green" title="E56">A171</font></i></a></i>;<br/>
<a NAME="E4:61_10_1_2_3_3"/>
<font color="Maroon" title="c2">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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2_3_3"><i><font color="Green" title="E63">A183</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_2_3_3"><i><font color="Green" title="E64">A184</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:61_10_1_2_3_3"/><b>then </b>
<font color="Maroon" title="c2">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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E6:61_10_1_2_3_3"/><b>then </b>
<font color="Maroon" title="c2">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">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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/>
<a NAME="E7:61_10_1_2_3_3"/><b>then </b>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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>hence </b><a NAME="E8:61_10_1_2_3_3"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:61_10_1_2_3"/><i><font color="Green" title="E63">A185</font></i>: 
<font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_3_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_3_4"/><i><font color="Green" title="E64">A186</font></i>: 
not <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_10_1_2_3_4"/><i><font color="Green" title="E65">A187</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_2"><i><font color="Green" title="E58">A173</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E5:61_10_1_2_3_4"/><i><font color="Green" title="E66">A188</font></i>: 
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">y2</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_2"><i><font color="Green" title="E56">A171</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:61_10_1_2_3_4"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2_3_4"><i><font color="Green" title="E64">A186</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_2_3_4"><i><font color="Green" title="E65">A187</font></i></a>, <a class="txt" href="neckla_3.html#E5:61_10_1_2_3_4"><i><font color="Green" title="E66">A188</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:61_10_1_2_3"/><i><font color="Green" title="E64">A189</font></i>: 
<font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_3_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_3_5"/><i><font color="Green" title="E65">A190</font></i>: 
not <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_10_1_2_3_5"/><i><font color="Green" title="E66">A191</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_2"><i><font color="Green" title="E58">A173</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E5:61_10_1_2_3_5"/><i><font color="Green" title="E67">A192</font></i>: 
( <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">y2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_2"><i><font color="Green" title="E57">A172</font></i></a></i>;<br/>
<a NAME="E6:61_10_1_2_3_5"/>
<font color="Maroon" title="c28">y1</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="c3">R1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2_3_5"><i><font color="Green" title="E65">A190</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_2_3_5"><i><font color="Green" title="E66">A191</font></i></a>, <a class="txt" href="neckla_3.html#E5:61_10_1_2_3_5"><i><font color="Green" title="E67">A192</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="E7:61_10_1_2_3_5"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</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="E6:61_10_1_2_3"/>
<font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_3_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_3_6"/><i><font color="Green" title="E65">A193</font></i>: 
not <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E3:61_10_1_2_3_6"/><i><font color="Green" title="E66">A194</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_2"><i><font color="Green" title="E58">A173</font></i></a></i>;<br/>
<a NAME="E4:61_10_1_2_3_6"/>
<font color="Maroon" title="c2">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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2_3_6"><i><font color="Green" title="E65">A193</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_2_3_6"><i><font color="Green" title="E66">A194</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:61_10_1_2_3_6"/><b>then </b>
<font color="Maroon" title="c2">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="c11">R</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E6:61_10_1_2_3_6"/><b>then </b>
<font color="Maroon" title="c2">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">G</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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/>
<a NAME="E7:61_10_1_2_3_6"/><b>then </b>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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>hence </b><a NAME="E8:61_10_1_2_3_6"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:61_10_1_2_3"/>
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font> &amp; <font color="Maroon" title="c25">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">u</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">v</font> &amp; <font color="Maroon" title="c27">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61_10_1_2_3"><i><font color="Green" title="E60">A175</font></i></a>, <a class="txt" href="neckla_3.html#E2:61_10_1_2_3"><i><font color="Green" title="E61">A179</font></i></a>, <a class="txt" href="neckla_3.html#E3:61_10_1_2_3"><i><font color="Green" title="E62">A182</font></i></a>, <a class="txt" href="neckla_3.html#E4:61_10_1_2_3"><i><font color="Green" title="E63">A185</font></i></a>, <a class="txt" href="neckla_3.html#E5:61_10_1_2_3"><i><font color="Green" title="E64">A189</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:61_10_1_2"/><b>then </b><i><font color="Green" title="E60">A195</font></i>: 
<font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c25">v</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c27">w</font> <a href="incproj.html#R2" title="INCPROJ:pred.2">are_mutually_different</a> 
 <b>by </b><i><a class="ref" href="incproj.html#D6" title="INCPROJ:def.6">INCPROJ:def 6</a></i>;<br/><a NAME="E18:61_10_1_2"/><i><font color="Green" title="E61">A196</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">v</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_4"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E2:61_10_1_2_4"/><i><font color="Green" title="E62">A197</font></i>: 
( <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_2"><i><font color="Green" title="E57">A172</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:61_10_1_2_4"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">v</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:61_10_1_2_4"><i><font color="Green" title="E62">A197</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:61_10_1_2"/><i><font color="Green" title="E62">A198</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E2:61_10_1_2_5"/><i><font color="Green" title="E63">A199</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_2"><i><font color="Green" title="E58">A173</font></i></a></i>;<br/>
<a NAME="E3:61_10_1_2_5"/>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="necklace.html#D4" title="NECKLACE:def.4">NECKLACE:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:61_10_1_2_5"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:61_10_1_2_5"><i><font color="Green" title="E63">A199</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:61_10_1_2"/><i><font color="Green" title="E63">A200</font></i>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_6"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E2:61_10_1_2_6"/><i><font color="Green" title="E64">A201</font></i>: 
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_2"><i><font color="Green" title="E56">A171</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:61_10_1_2_6"/>
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E2:61_10_1_2_6"><i><font color="Green" title="E64">A201</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:61_10_1_2"/><i><font color="Green" title="E64">A202</font></i>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">u</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_7"/><i><font color="Green" title="E65">A203</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">u</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_10_1_2_7"/><i><font color="Green" title="E66">A204</font></i>: 
( <font color="Maroon" title="c24">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E8:61_10_1_2"><i><font color="Green" title="E56">A171</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E5:61_10_1_2_7"/><i><font color="Green" title="E67">A205</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">y2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_2"><i><font color="Green" title="E58">A173</font></i></a></i>;<br/>
<a NAME="E6:61_10_1_2_7"/>
<font color="Maroon" title="c24">u</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:61_10_1_2_7"><i><font color="Green" title="E66">A204</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">u</font> = <font color="Maroon" title="c24">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E8:61_10_1_2_7"/>
<font color="Maroon" title="c27">w</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:61_10_1_2_7"><i><font color="Green" title="E67">A205</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c31">w</font> = <font color="Maroon" title="c27">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E10:61_10_1_2_7"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:61_10_1_2_7"/><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="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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#E1:61_10_1_2_7"><i><font color="Green" title="E65">A203</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="E12:61_10_1_2_7"/><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="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>
 
;<br/>
<a NAME="E13:61_10_1_2_7"/><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="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c11">R</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>;<br/>
<a NAME="E14:61_10_1_2_7"/><b>then </b><i><font color="Green" title="E68">A206</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font> <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 <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_10_1_2_7_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">R2</font> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E14:61_10_1_2_7"><i><font color="Green" title="E68">A206</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="61_10_1_2_7_1_1"><b>suppose </b></a><a NAME="E1:61_10_1_2_7_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font>
 ;<br/><div class="add"><a NAME="E2:61_10_1_2_7_1_1"/><b>then </b>
( <font color="Maroon" title="c30">u</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="c3">R1</font> &amp; <font color="Maroon" title="c31">w</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="c3">R1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:61_10_1_2_7_1_1"/><b>then </b>
<font color="Maroon" title="c30">u</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="c3">R1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_10_1_2_7"><i><font color="Green" title="E66">A204</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="E4:61_10_1_2_7_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</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="61_10_1_2_7_1_2"><b>suppose </b></a><a NAME="E1:61_10_1_2_7_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">u</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">R2</font>
 ;<br/><div class="add"><a NAME="E2:61_10_1_2_7_1_2"/><b>then </b>
( <font color="Maroon" title="c30">u</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="c4">R2</font> &amp; <font color="Maroon" title="c31">w</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="c4">R2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:61_10_1_2_7_1_2"/><b>then </b>
<font color="Maroon" title="c31">w</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="c3">R1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:61_10_1_2_7"><i><font color="Green" title="E67">A205</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="E4:61_10_1_2_7_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</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><b>end;</b></div>

</div><b>end;</b></div><a NAME="E22:61_10_1_2"/>
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">v</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_8"/><i><font color="Green" title="E65">A207</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c25">v</font>,<font color="Maroon" title="c27">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c28">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">R2</font><b> such that </b><br/><a NAME="E3:61_10_1_2_8"/><i><font color="Green" title="E66">A208</font></i>: 
( <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y1</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c28">y1</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E9:61_10_1_2"><i><font color="Green" title="E57">A172</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R1</font><b> such that </b><br/><a NAME="E5:61_10_1_2_8"/><i><font color="Green" title="E67">A209</font></i>: 
( <font color="Maroon" title="c27">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">y2</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c29">y2</font>,<font color="Maroon" title="c2">x</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="neckla_3.html#E10:61_10_1_2"><i><font color="Green" title="E58">A173</font></i></a></i>;<br/>
<a NAME="E6:61_10_1_2_8"/>
<font color="Maroon" title="c25">v</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:61_10_1_2_8"><i><font color="Green" title="E66">A208</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">v</font> = <font color="Maroon" title="c25">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E8:61_10_1_2_8"/>
<font color="Maroon" title="c27">w</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="c3">R1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E5:61_10_1_2_8"><i><font color="Green" title="E67">A209</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c31">w</font> = <font color="Maroon" title="c27">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">R</font> <b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</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="E10:61_10_1_2_8"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:61_10_1_2_8"/><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="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</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#E1:61_10_1_2_8"><i><font color="Green" title="E65">A207</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="E12:61_10_1_2_8"/><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="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">G</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">R</font>
 
;<br/>
<a NAME="E13:61_10_1_2_8"/><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="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c11">R</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></i>;<br/>
<a NAME="E14:61_10_1_2_8"/><b>then </b><i><font color="Green" title="E68">A210</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font> <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 <font color="Maroon" title="c4">R2</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E12:61"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_10_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_10_1_2_8_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">R2</font> )
 </span><b>by </b><i><a class="txt" href="neckla_3.html#E14:61_10_1_2_8"><i><font color="Green" title="E68">A210</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="61_10_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:61_10_1_2_8_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R1</font>
 ;<br/><div class="add"><a NAME="E2:61_10_1_2_8_1_1"/><b>then </b>
( <font color="Maroon" title="c30">v</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="c3">R1</font> &amp; <font color="Maroon" title="c31">w</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="c3">R1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:61_10_1_2_8_1_1"/><b>then </b>
<font color="Maroon" title="c30">v</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="c3">R1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E3:61_10_1_2_8"><i><font color="Green" title="E66">A208</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="E4:61_10_1_2_8_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</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="61_10_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:61_10_1_2_8_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">v</font>,<font color="Maroon" title="c31">w</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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">R2</font>
 ;<br/><div class="add"><a NAME="E2:61_10_1_2_8_1_2"/><b>then </b>
( <font color="Maroon" title="c30">v</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="c4">R2</font> &amp; <font color="Maroon" title="c31">w</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="c4">R2</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:61_10_1_2_8_1_2"/><b>then </b>
<font color="Maroon" title="c31">w</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="c3">R1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R2</font>
 <b>by </b><i><a class="txt" href="neckla_3.html#E5:61_10_1_2_8"><i><font color="Green" title="E67">A209</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="E4:61_10_1_2_8_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E1:61"><i><font color="Green" title="E37">A1</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><b>end;</b></div>

</div><b>end;</b></div><a NAME="E23:61_10_1_2"/><b>then </b><i><font color="Green" title="E65">A211</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c23">Z</font> <a href="necklace.html#R1" title="NECKLACE:pred.1">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4
 <b>by </b><i><a class="txt" href="neckla_3.html#E4:61_10_1_2"><i><font color="Green" title="E55">A163</font></i></a>, <a class="txt" href="neckla_3.html#E17:61_10_1_2"><i><font color="Green" title="E60">A195</font></i></a>, <a class="txt" href="neckla_3.html#E18:61_10_1_2"><i><font color="Green" title="E61">A196</font></i></a>, <a class="txt" href="neckla_3.html#E19:61_10_1_2"><i><font color="Green" title="E62">A198</font></i></a>, <a class="txt" href="neckla_3.html#E20:61_10_1_2"><i><font color="Green" title="E63">A200</font></i></a>, <a class="txt" href="neckla_3.html#E21:61_10_1_2"><i><font color="Green" title="E64">A202</font></i></a>, <a class="ref" href="neckla_3.html#T39" target="_self" title="NECKLA_3:th.39">Th39</a></i>;<br/><a NAME="E24:61_10_1_2"/>
not  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c23">Z</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_9"><b>proof </b></a><div class="add">

<a NAME="E1:61_10_1_2_9"/>
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="c23">Z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Z</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="E2:61_10_1_2_9"/><b>then </b>
<font color="Maroon" title="c24">u</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c23">Z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:61_10_1_2_9"/>
not  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c23">Z</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c28">H</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c23">Z</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E26:61_10_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="necklace.html#R2" title="NECKLACE:pred.2">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_10_1_2_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_10_1_2_10"/>
not <font color="Maroon" title="c1">G</font> <a href="necklace.html#R2" title="NECKLACE:pred.2">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4
 ;<br/>

<a NAME="E2:61_10_1_2_10"/><b>then </b>
<font color="Maroon" title="c1">G</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 
<b>by </b><i><a class="ref" href="neckla_2.html#D1" title="NECKLA_2:def.1">NECKLA_2:def 1</a></i>;<br/>
<a NAME="E3:61_10_1_2_10"/><b>then </b>
<font color="Maroon" title="c28">H</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 
<b>by </b><i><a class="ref" href="neckla_3.html#T23" target="_self" title="NECKLA_3:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E4:61_10_1_2_10"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E23:61_10_1_2"><i><font color="Green" title="E65">A211</font></i></a>, <a class="ref" href="neckla_2.html#D1" title="NECKLA_2:def.1">NECKLA_2:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E27:61_10_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="necklace.html#R2" title="NECKLACE:pred.2">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div>
