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

<b>set </b><font color="Maroon" title="c1">S</font> =  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4;<br/>
<b>set </b><font color="Maroon" title="c2">T</font> =  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c3">g</font> = 0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3;<br/>
<b>set </b><font color="Maroon" title="c4">h</font> = 2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2;<br/>
<b>set </b><font color="Maroon" title="c5">f</font> = <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span>;<br/>
<i><font color="Green" title="E30">A1</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,3</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T45" title="ENUMSET1:th.45">ENUMSET1:45</a></i>
;<br/>
<a NAME="E2:54"/><b>then </b><i><font color="Green" title="E31">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<b>by </b><i><a class="ref" href="necklace.html#T2" target="_self" title="NECKLACE:th.2">Th2</a>, <a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a></i>;<br/>
<a NAME="E3:54"/><i><font color="Green" title="E32">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,3</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T67" title="FUNCT_4:th.67">FUNCT_4:67</a></i>;<br/>
<a NAME="E4:54"/><i><font color="Green" title="E33">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T67" title="FUNCT_4:th.67">FUNCT_4:67</a></i>;<br/>
<a NAME="E5:54"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span></span>)</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 <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</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="E1:54_2"/><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:54_2"/><i><font color="Green" title="E35">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></i>;<br/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span></span>)</span> = 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default">1,3,0,2</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
<b>by </b><i><a class="txt" href="necklace.html#E3:54"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="necklace.html#E4:54"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="enumset1.html#T45" title="ENUMSET1:th.45">ENUMSET1:45</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T112" title="ENUMSET1:th.112">ENUMSET1:112</a></i>
;<br/>
<a NAME="E4:54_2"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 4
 
<b>by </b><i><a class="txt" href="necklace.html#E1:54_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="necklace.html#E2:54_2"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="necklace.html#T2" target="_self" title="NECKLACE:th.2">Th2</a></i>;<br/>
<a NAME="E5:54_2"/><b>then </b>
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a></i>;<br/>
<b>hence </b><a NAME="E6:54_2"/>
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">f</font> = <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="necklace.html#E2:54"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_3"/>
( ( not  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4 is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp; not  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> ) or  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4 is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> or  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1"><b>case </b></a><b>that </b><a NAME="E1:54_3_1"/>
not  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4 is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <b>and </b><br/><a NAME="E2:54_3_1"/>
not  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 ;<br/><div class="add"><a NAME="E3:54_3_1"/><i><font color="Green" title="E34">A7</font></i>: 
0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3 is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:54_3_1_1"/><i><font color="Green" title="E35">A8</font></i>: 
0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 3</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D4" title="FUNCT_4:def.4">FUNCT_4:def 4</a></i>;<br/>
<a NAME="E2:54_3_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">1 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 3</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_3_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">1 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 3</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:54_3_1_1_1"/><i><font color="Green" title="E36">A9</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 1</span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">1 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 3</span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:54_3_1_1_1"/><i><font color="Green" title="E37">A10</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E5:54_3_1_1_1"/><i><font color="Green" title="E38">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">1 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">3</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E6:54_3_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="necklace.html#E3:54_3_1_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="necklace.html#E4:54_3_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:54_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="necklace.html#E3:54_3_1_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="necklace.html#E5:54_3_1_1_1"><i><font color="Green" title="E38">A11</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>
<b>hence </b><a NAME="E3:54_3_1_1"/>
0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3 is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="necklace.html#E1:54_3_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="knaster.html#T1" title="KNASTER:th.1">KNASTER:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:54_3_1"/><i><font color="Green" title="E35">A12</font></i>: 
2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2 is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:54_3_1_2"/><i><font color="Green" title="E36">A13</font></i>: 
2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 0</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">3</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D4" title="FUNCT_4:def.4">FUNCT_4:def 4</a></i>;<br/>
<a NAME="E2:54_3_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 0</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">3 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_3_1_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 0</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">3 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 2</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:54_3_1_2_1"/><i><font color="Green" title="E37">A14</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 0</span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">3 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:54_3_1_2_1"/><i><font color="Green" title="E38">A15</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E5:54_3_1_2_1"/><i><font color="Green" title="E39">A16</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">3 <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E6:54_3_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E3:54_3_1_2_1"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="necklace.html#E4:54_3_1_2_1"><i><font color="Green" title="E38">A15</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:54_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="necklace.html#E3:54_3_1_2_1"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="necklace.html#E5:54_3_1_2_1"><i><font color="Green" title="E39">A16</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>
<b>hence </b><a NAME="E3:54_3_1_2"/>
2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2 is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="necklace.html#E1:54_3_1_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="knaster.html#T1" title="KNASTER:th.1">KNASTER:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:54_3_1"/><i><font color="Green" title="E36">A17</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_3_1_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:54_3_1_3"/><i><font color="Green" title="E37">A18</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:54_3_1_3"/>
( ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 
<b>by </b><i><a class="txt" href="necklace.html#E3:54_3_1_3"><i><font color="Green" title="E37">A18</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:54_3_1_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E6:54_3_1"/><i><font color="Green" title="E37">A19</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_3_1_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:54_3_1_4"/><i><font color="Green" title="E38">A20</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:54_3_1_4"/>
( ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp; ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="necklace.html#E3:54"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="necklace.html#E4:54"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="necklace.html#E3:54_3_1_4"><i><font color="Green" title="E38">A20</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:54_3_1_4"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:54_3_1"/><i><font color="Green" title="E38">A21</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="necklace.html#E3:54_3_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="necklace.html#E4:54_3_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="knaster.html#T1" title="KNASTER:th.1">KNASTER:1</a></i>;<br/><b>set </b><font color="Maroon" title="c7">A</font> = 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> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c8">B</font> = <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>;<br/><a NAME="E8:54_3_1"/><i><font color="Green" title="E39">A22</font></i>: 
<span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</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="E1:54_3_1_5"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 ;<br/>

<a NAME="E2:54_3_1_5"/><b>then </b><i><font color="Green" title="E40">A23</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">C</font> = 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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>;<br/>
<a NAME="E3:54_3_1_5"/>
( 0 <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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> &amp; 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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> &amp; 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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> &amp; 3 <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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i><a class="txt" href="necklace.html#E1:54"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="necklace.html#E2:54"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">x'</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="txt" href="necklace.html#E2:54_3_1_5"><i><font color="Green" title="E40">A23</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:54_3_1_5"/><i><font color="Green" title="E41">A24</font></i>: 
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:54_3_1_5_1"/>
not <font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_5_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_3_1_5_1_1"/>
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:54_3_1_5_1_1"/>
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4
 <b>and </b><br/><a NAME="E4:54_3_1_5_1_1"/><i><font color="Green" title="E42">A25</font></i>: 
( <font color="Maroon" title="c11">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c11">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="necklace.html#T20" target="_self" title="NECKLACE:th.20">Th20</a></i>;<br/>
<a NAME="E5:54_3_1_5_1_1"/>
( ( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 
<b>by </b><i><a class="txt" href="necklace.html#E2:54_3_1_5"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="necklace.html#E4:54_3_1_5_1_1"><i><font color="Green" title="E42">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:54_3_1_5_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:54_3_1_5_1"/>
<font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:54_3_1_5"/>
not <font color="Maroon" title="c9">x</font> <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 <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E2:54_3_1_5"><i><font color="Green" title="E40">A23</font></i></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/>
<a NAME="E7:54_3_1_5"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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 <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1_5"><i><font color="Green" title="E41">A24</font></i></a>, <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="E8:54_3_1_5"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E9:54_3_1"/>
<font color="Maroon" title="c6">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_3.html#D5" title="ORDERS_3:def.5">ORDERS_3:def 5</a><br/>

<b>assume </b><a NAME="E1:54_3_1_6"/><i><font color="Green" title="E40">A26</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">y</font>
 ;<br/>

<a NAME="E2:54_3_1_6"/>
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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">i</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c12">j</font> = <font color="Maroon" title="c10">y</font> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="necklace.html#T4" target="_self" title="NECKLACE:th.4">Th4</a></i>;<br/>
<a NAME="E4:54_3_1_6"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E1:54_3_1_6"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E5:54_3_1_6"/><b>then </b><i><font color="Green" title="E41">A27</font></i>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 or <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="necklace.html#T25" target="_self" title="NECKLACE:th.25">Th25</a></i>;<br/>
<b>let </b><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font> be   <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> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E6:54_3_1_6"/><i><font color="Green" title="E42">A28</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> )
 ;<br/>

<a NAME="E7:54_3_1_6"/><i><font color="Green" title="E43">A29</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>;<br/>
<a NAME="E8:54_3_1_6"/><i><font color="Green" title="E44">A30</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,3</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>;<br/>
<a NAME="E9:54_3_1_6"/><i><font color="Green" title="E45">A31</font></i>: 
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E7:54_3_1_6"><i><font color="Green" title="E43">A29</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E10:54_3_1_6"/><i><font color="Green" title="E46">A32</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E7:54_3_1_6"><i><font color="Green" title="E43">A29</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E11:54_3_1_6"/><i><font color="Green" title="E47">A33</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E8:54_3_1_6"><i><font color="Green" title="E44">A30</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E12:54_3_1_6"/><i><font color="Green" title="E48">A34</font></i>: 
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E8:54_3_1_6"><i><font color="Green" title="E44">A30</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<i><font color="Green" title="E49">A35</font></i>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 = 
<span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
<b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="necklace.html#E9:54_3_1_6"><i><font color="Green" title="E45">A31</font></i></a>, <a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>
;<br/>
<i><font color="Green" title="E50">A36</font></i>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="necklace.html#E10:54_3_1_6"><i><font color="Green" title="E46">A32</font></i></a>, <a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></i>
<br/>.= 
3
<b>by </b><i><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>
;<br/>
<i><font color="Green" title="E51">A37</font></i>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="necklace.html#E11:54_3_1_6"><i><font color="Green" title="E47">A33</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>
;<br/>
<i><font color="Green" title="E52">A38</font></i>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 = 
<span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<b>by </b><i><a class="txt" href="necklace.html#E12:54_3_1_6"><i><font color="Green" title="E48">A34</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_3_1_6_5"/>
( ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 </span><b>by </b><i><a class="txt" href="necklace.html#E1:54"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="necklace.html#E2:54"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_1"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_1"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_6_5_1"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1_6"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_2"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_2"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_6_5_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <b>by </b><i><a class="txt" href="necklace.html#E6:54_3_1_6"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="necklace.html#E13:54_3_1_6"><i><font color="Green" title="E49">A35</font></i></a>, <a class="txt" href="necklace.html#E14:54_3_1_6"><i><font color="Green" title="E50">A36</font></i></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:54_3_1_6_5_2"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E8:54_3_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_3"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_3"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_6_5_3"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1_6"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_4"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_4"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_6_5_4"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1_6"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_5"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_5"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_6_5_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <b>by </b><i><a class="txt" href="necklace.html#E6:54_3_1_6"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="necklace.html#E13:54_3_1_6"><i><font color="Green" title="E49">A35</font></i></a>, <a class="txt" href="necklace.html#E14:54_3_1_6"><i><font color="Green" title="E50">A36</font></i></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:54_3_1_6_5_5"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E8:54_3_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_6"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_6"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_6_5_6"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1_6"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_7"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_7"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_6_5_7"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <b>by </b><i><a class="txt" href="necklace.html#E6:54_3_1_6"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="necklace.html#E14:54_3_1_6"><i><font color="Green" title="E50">A36</font></i></a>, <a class="txt" href="necklace.html#E15:54_3_1_6"><i><font color="Green" title="E51">A37</font></i></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:54_3_1_6_5_7"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E8:54_3_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_8"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_8"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_6_5_8"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1_6"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_9"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_9"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_6_5_9"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1_6"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_10"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_10"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_6_5_10"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <b>by </b><i><a class="txt" href="necklace.html#E6:54_3_1_6"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="necklace.html#E14:54_3_1_6"><i><font color="Green" title="E50">A36</font></i></a>, <a class="txt" href="necklace.html#E15:54_3_1_6"><i><font color="Green" title="E51">A37</font></i></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:54_3_1_6_5_10"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E8:54_3_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_11"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_11"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_6_5_11"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1_6"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_12"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_12"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_6_5_12"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <b>by </b><i><a class="txt" href="necklace.html#E6:54_3_1_6"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="necklace.html#E15:54_3_1_6"><i><font color="Green" title="E51">A37</font></i></a>, <a class="txt" href="necklace.html#E16:54_3_1_6"><i><font color="Green" title="E52">A38</font></i></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:54_3_1_6_5_12"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E8:54_3_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_13"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_13"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_6_5_13"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1_6"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_14"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_14"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_6_5_14"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1_6"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_15"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_15"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_6_5_15"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <b>by </b><i><a class="txt" href="necklace.html#E6:54_3_1_6"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="necklace.html#E15:54_3_1_6"><i><font color="Green" title="E51">A37</font></i></a>, <a class="txt" href="necklace.html#E16:54_3_1_6"><i><font color="Green" title="E52">A38</font></i></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:54_3_1_6_5_15"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E8:54_3_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_6_5_16"><b>suppose </b></a><a NAME="E1:54_3_1_6_5_16"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_6_5_16"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1_6"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="necklace.html#E5:54_3_1"><i><font color="Green" title="E36">A17</font></i></a>, <a class="ref" href="necklace.html#T7" target="_self" title="NECKLACE:th.7">Th7</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default">1,3,0,2</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
<b>by </b><i><a class="txt" href="necklace.html#E3:54"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="necklace.html#E4:54"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="enumset1.html#T45" title="ENUMSET1:th.45">ENUMSET1:45</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T112" title="ENUMSET1:th.112">ENUMSET1:112</a></i>
<br/>.= 
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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="necklace.html#T2" target="_self" title="NECKLACE:th.2">Th2</a>, <a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a></i>
<br/>.= 
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> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <b>by </b><i><a class="txt" href="necklace.html#E7:54_3_1"><i><font color="Green" title="E38">A21</font></i></a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></i>;<br/><b>take </b>
<font color="Maroon" title="c9">F</font>
;<br/><b>thus </b><a NAME="E12:54_3_1"/>
<font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 ;<br/><b>thus </b><a NAME="E13:54_3_1"/>
<font color="Maroon" title="c9">F</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <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> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_3.html#D5" title="ORDERS_3:def.5">ORDERS_3:def 5</a><br/>

<b>assume </b><a NAME="E1:54_3_1_8"/><i><font color="Green" title="E40">A39</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">y</font>
 ;<br/>

<i><font color="Green" title="E41">A40</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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span> = 
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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></i>
<br/>.= 
4
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a></i>
;<br/>
<a NAME="E3:54_3_1_8"/><b>then </b>
( <font color="Maroon" title="c10">x</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> &amp; <font color="Maroon" title="c11">y</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> )
 
<b>by </b><i><a class="ref" href="necklace.html#T4" target="_self" title="NECKLACE:th.4">Th4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c11">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:54_3_1_8"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</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> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E1:54_3_1_8"><i><font color="Green" title="E40">A39</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E6:54_3_1_8"/><b>then </b><i><font color="Green" title="E42">A41</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</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 <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></i>;<br/>
<a NAME="E7:54_3_1_8"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:54_3_1_8"/><b>then </b>
( not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">k</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b1">l</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 </span>}</span>  &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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 <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> )
 
<b>by </b><i><a class="txt" href="necklace.html#E6:54_3_1_8"><i><font color="Green" title="E42">A41</font></i></a>, <a class="ref" href="necklace.html#T19" target="_self" title="NECKLACE:th.19">Th19</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E9:54_3_1_8"/><b>then </b><i><font color="Green" title="E43">A42</font></i>: 
( not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">k</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 </span>}</span>  &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E10:54_3_1_8"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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 <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E6:54_3_1_8"><i><font color="Green" title="E42">A41</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="E11:54_3_1_8"/><b>then </b><i><font color="Green" title="E44">A43</font></i>: 
(  not <font color="Maroon" title="c10">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 <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y</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="E12:54_3_1_8"/><i><font color="Green" title="E45">A44</font></i>: 
( <font color="Maroon" title="c10">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 <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span> )
 
;<br/>
<a NAME="E13:54_3_1_8"/><i><font color="Green" title="E46">A45</font></i>: 
( <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">j</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_8_2"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_3_1_8_2_1"/><i><font color="Green" title="E47">A46</font></i>: 
( <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> or <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_3_1_8_2_1_1"/>
( <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> or <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> )
 </span><b>by </b><i><a class="txt" href="necklace.html#E1:54_3_1_8_2_1"><i><font color="Green" title="E47">A46</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_2_1_1_1"><b>suppose </b></a><a NAME="E1:54_3_1_8_2_1_1_1"/><i><font color="Green" title="E48">A47</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font>
 ;<br/><div class="add"><a NAME="E2:54_3_1_8_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4
 <b>by </b><i><a class="txt" href="necklace.html#E2:54_3_1_8"><i><font color="Green" title="E41">A40</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/><b>hence </b><a NAME="E3:54_3_1_8_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="necklace.html#E9:54_3_1_8"><i><font color="Green" title="E43">A42</font></i></a>, <a class="txt" href="necklace.html#E1:54_3_1_8_2_1_1_1"><i><font color="Green" title="E48">A47</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_2_1_1_2"><b>suppose </b></a><a NAME="E1:54_3_1_8_2_1_1_2"/><i><font color="Green" title="E48">A48</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font>
 ;<br/><div class="add"><a NAME="E2:54_3_1_8_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4
 <b>by </b><i><a class="txt" href="necklace.html#E2:54_3_1_8"><i><font color="Green" title="E41">A40</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/><b>hence </b><a NAME="E3:54_3_1_8_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="necklace.html#E9:54_3_1_8"><i><font color="Green" title="E43">A42</font></i></a>, <a class="txt" href="necklace.html#E1:54_3_1_8_2_1_1_2"><i><font color="Green" title="E48">A48</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:54_3_1_8_2"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="necklace.html#E11:54_3_1_8"><i><font color="Green" title="E44">A43</font></i></a>, <a class="txt" href="necklace.html#E12:54_3_1_8"><i><font color="Green" title="E45">A44</font></i></a>, <a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>;<br/>

<b>assume </b><a NAME="E14:54_3_1_8"/><i><font color="Green" title="E47">A49</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font> )
 ;<br/>

<a NAME="E15:54_3_1_8"/><i><font color="Green" title="E48">A50</font></i>: 
<span class="p1">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,1
 
<b>by </b><i><a class="ref" href="necklace.html#T11" target="_self" title="NECKLACE:th.11">Th11</a></i>;<br/>
<a NAME="E16:54_3_1_8"/><i><font color="Green" title="E49">A51</font></i>: 
<span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0,2 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 2,3
 
<b>by </b><i><a class="ref" href="necklace.html#T11" target="_self" title="NECKLACE:th.11">Th11</a></i>;<br/>
<a NAME="E17:54_3_1_8"/><i><font color="Green" title="E50">A52</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,3</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="necklace.html#E15:54_3_1_8"><i><font color="Green" title="E48">A50</font></i></a>, <a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>;<br/>
<a NAME="E18:54_3_1_8"/><i><font color="Green" title="E51">A53</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="necklace.html#E16:54_3_1_8"><i><font color="Green" title="E49">A51</font></i></a>, <a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>;<br/>
<a NAME="E19:54_3_1_8"/><i><font color="Green" title="E52">A54</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3_1_8_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_3_1_8_3"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:54_3_1_8_3"/><i><font color="Green" title="E53">A55</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> &amp; <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:54_3_1_8_3"/>
( ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp; ( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="necklace.html#E17:54_3_1_8"><i><font color="Green" title="E50">A52</font></i></a>, <a class="txt" href="necklace.html#E18:54_3_1_8"><i><font color="Green" title="E51">A53</font></i></a>, <a class="txt" href="necklace.html#E3:54_3_1_8_3"><i><font color="Green" title="E53">A55</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:54_3_1_8_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E20:54_3_1_8"/><i><font color="Green" title="E53">A56</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E17:54_3_1_8"><i><font color="Green" title="E50">A52</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E21:54_3_1_8"/><i><font color="Green" title="E54">A57</font></i>: 
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E17:54_3_1_8"><i><font color="Green" title="E50">A52</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E22:54_3_1_8"/><i><font color="Green" title="E55">A58</font></i>: 
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E18:54_3_1_8"><i><font color="Green" title="E51">A53</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E23:54_3_1_8"/><i><font color="Green" title="E56">A59</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E18:54_3_1_8"><i><font color="Green" title="E51">A53</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<i><font color="Green" title="E57">A60</font></i>: <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="necklace.html#E3:54_3_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="necklace.html#E4:54_3_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="necklace.html#E5:54_3_1"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="necklace.html#E6:54_3_1"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="necklace.html#T8" target="_self" title="NECKLACE:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="necklace.html#E19:54_3_1_8"><i><font color="Green" title="E52">A54</font></i></a>, <a class="txt" href="necklace.html#E20:54_3_1_8"><i><font color="Green" title="E53">A56</font></i></a>, <a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="necklace.html#E15:54_3_1_8"><i><font color="Green" title="E48">A50</font></i></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>
;<br/>
<i><font color="Green" title="E58">A61</font></i>: <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<b>by </b><i><a class="txt" href="necklace.html#E3:54_3_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="necklace.html#E4:54_3_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="necklace.html#E5:54_3_1"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="necklace.html#E6:54_3_1"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="necklace.html#T8" target="_self" title="NECKLACE:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<b>by </b><i><a class="txt" href="necklace.html#E19:54_3_1_8"><i><font color="Green" title="E52">A54</font></i></a>, <a class="txt" href="necklace.html#E21:54_3_1_8"><i><font color="Green" title="E54">A57</font></i></a>, <a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="necklace.html#E15:54_3_1_8"><i><font color="Green" title="E48">A50</font></i></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>
;<br/>
<i><font color="Green" title="E59">A62</font></i>: <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
<b>by </b><i><a class="txt" href="necklace.html#E3:54_3_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="necklace.html#E4:54_3_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="necklace.html#E5:54_3_1"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="necklace.html#E6:54_3_1"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="necklace.html#T8" target="_self" title="NECKLACE:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
<b>by </b><i><a class="txt" href="necklace.html#E22:54_3_1_8"><i><font color="Green" title="E55">A58</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
2
<b>by </b><i><a class="txt" href="necklace.html#E16:54_3_1_8"><i><font color="Green" title="E49">A51</font></i></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>
;<br/>
<i><font color="Green" title="E60">A63</font></i>: <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="necklace.html#E3:54_3_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="necklace.html#E4:54_3_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="necklace.html#E5:54_3_1"><i><font color="Green" title="E36">A17</font></i></a>, <a class="txt" href="necklace.html#E6:54_3_1"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="necklace.html#T8" target="_self" title="NECKLACE:th.8">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 0,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="necklace.html#E23:54_3_1_8"><i><font color="Green" title="E56">A59</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
3
<b>by </b><i><a class="txt" href="necklace.html#E16:54_3_1_8"><i><font color="Green" title="E49">A51</font></i></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_3_1_8_8"/>
( ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 </span><b>by </b><i><a class="txt" href="necklace.html#E2:54_3_1_8"><i><font color="Green" title="E41">A40</font></i></a>, <a class="ref" href="necklace.html#T2" target="_self" title="NECKLACE:th.2">Th2</a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_1"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_8_8_1"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E13:54_3_1_8"><i><font color="Green" title="E46">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_2"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_2"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_8_8_2"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E13:54_3_1_8"><i><font color="Green" title="E46">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_3"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_3"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_8_8_3"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="necklace.html#E14:54_3_1_8"><i><font color="Green" title="E47">A49</font></i></a>, <a class="txt" href="necklace.html#E26:54_3_1_8"><i><font color="Green" title="E59">A62</font></i></a>, <a class="txt" href="necklace.html#E27:54_3_1_8"><i><font color="Green" title="E60">A63</font></i></a></i>;<br/><a NAME="E3:54_3_1_8_8_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E4:54_3_1_8_8_3"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_4"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_4"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_8_8_4"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="necklace.html#E14:54_3_1_8"><i><font color="Green" title="E47">A49</font></i></a>, <a class="txt" href="necklace.html#E25:54_3_1_8"><i><font color="Green" title="E58">A61</font></i></a>, <a class="txt" href="necklace.html#E26:54_3_1_8"><i><font color="Green" title="E59">A62</font></i></a></i>;<br/><a NAME="E3:54_3_1_8_8_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E4:54_3_1_8_8_4"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_5"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_5"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_8_8_5"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E13:54_3_1_8"><i><font color="Green" title="E46">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_6"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_6"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_8_8_6"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E13:54_3_1_8"><i><font color="Green" title="E46">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_7"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_7"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_8_8_7"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E13:54_3_1_8"><i><font color="Green" title="E46">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_8"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_8"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_8_8_8"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="necklace.html#E14:54_3_1_8"><i><font color="Green" title="E47">A49</font></i></a>, <a class="txt" href="necklace.html#E24:54_3_1_8"><i><font color="Green" title="E57">A60</font></i></a>, <a class="txt" href="necklace.html#E25:54_3_1_8"><i><font color="Green" title="E58">A61</font></i></a></i>;<br/><a NAME="E3:54_3_1_8_8_8"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E4:54_3_1_8_8_8"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_9"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_9"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_8_8_9"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="txt" href="necklace.html#E14:54_3_1_8"><i><font color="Green" title="E47">A49</font></i></a>, <a class="txt" href="necklace.html#E26:54_3_1_8"><i><font color="Green" title="E59">A62</font></i></a>, <a class="txt" href="necklace.html#E27:54_3_1_8"><i><font color="Green" title="E60">A63</font></i></a></i>;<br/><a NAME="E3:54_3_1_8_8_9"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E4:54_3_1_8_8_9"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_10"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_10"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_8_8_10"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E13:54_3_1_8"><i><font color="Green" title="E46">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_11"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_11"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_8_8_11"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E13:54_3_1_8"><i><font color="Green" title="E46">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_12"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_12"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_8_8_12"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E13:54_3_1_8"><i><font color="Green" title="E46">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_13"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_13"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_8_8_13"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="necklace.html#E14:54_3_1_8"><i><font color="Green" title="E47">A49</font></i></a>, <a class="txt" href="necklace.html#E25:54_3_1_8"><i><font color="Green" title="E58">A61</font></i></a>, <a class="txt" href="necklace.html#E26:54_3_1_8"><i><font color="Green" title="E59">A62</font></i></a></i>;<br/><a NAME="E3:54_3_1_8_8_13"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E4:54_3_1_8_8_13"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_14"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_14"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:54_3_1_8_8_14"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="necklace.html#E14:54_3_1_8"><i><font color="Green" title="E47">A49</font></i></a>, <a class="txt" href="necklace.html#E24:54_3_1_8"><i><font color="Green" title="E57">A60</font></i></a>, <a class="txt" href="necklace.html#E25:54_3_1_8"><i><font color="Green" title="E58">A61</font></i></a></i>;<br/><a NAME="E3:54_3_1_8_8_14"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E4:54_3_1_8_8_14"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_15"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_15"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_8_8_15"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E13:54_3_1_8"><i><font color="Green" title="E46">A45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_1_8_8_16"><b>suppose </b></a><a NAME="E1:54_3_1_8_8_16"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_1_8_8_16"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E13:54_3_1_8"><i><font color="Green" title="E46">A45</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_2"><b>case </b></a><a NAME="E1:54_3_2"/>
(  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4 is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> or  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_3_2"/>
(  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4 is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp;  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
