<?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> 1;<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> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c3">f</font> = 0 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> 0;<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">0 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> 0</span>)</span> = 
<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="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</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> 1</span>)</span>
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a></i>
;<br/>
<a NAME="E2:53"/><i><font color="Green" title="E31">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a></i>;<br/>
<a NAME="E3:53"/><i><font color="Green" title="E32">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 1</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#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 1</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="c4">g</font> = 0 <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> 0 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> 1</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> 1</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="necklace.html#E1:53"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="necklace.html#E2:53"><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#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green" title="E33">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <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="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E6:53"/><i><font color="Green" title="E34">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> <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="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E7:53"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c4">g</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="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">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> 1</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="E1:53_2"/><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font>
 ;<br/>

<a NAME="E2:53_2"/>
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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<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="c7">i</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c6">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="E4:53_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">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> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E1:53_2"><i><font color="Green" title="E36">A7</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="E5:53_2"/><b>then </b><i><font color="Green" title="E37">A8</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 or <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="c9">a</font>, <font color="Maroon" title="c10">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> 1</span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E6:53_2"/>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> )
 ;<br/>

<a NAME="E7:53_2"/><i><font color="Green" title="E38">A9</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#K4" title="NECKLACE:func.4">Necklace</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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a></i>;<br/>
<a NAME="E8:53_2"/><b>then </b><i><font color="Green" title="E39">A10</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:53_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E7:53_2"><i><font color="Green" title="E38">A9</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="E10:53_2"/>
<font color="Maroon" title="c9">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E5:53_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="necklace.html#E8:53_2"><i><font color="Green" title="E39">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:53"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>proof </b></a><div class="add">

<a NAME="E1:53_3"/>
for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</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> <font color="Maroon" title="c4">g</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> iff ( <font color="Olive" title="b2">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> <font color="Maroon" title="c4">g</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:53_3_1"/><i><font color="Green" title="E36">A11</font></i>: 
( <font color="Maroon" title="c6">y</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> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> implies ( <font color="Maroon" title="c5">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> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:53_3_1_1"/><i><font color="Green" title="E37">A12</font></i>: 
<font color="Maroon" title="c6">y</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> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E2:53_3_1_1"/><i><font color="Green" title="E38">A13</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
 ;<br/>

<a NAME="E3:53_3_1_1"/><i><font color="Green" title="E39">A14</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E6:53"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="necklace.html#E1:53_3_1_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:53_3_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E2:53_3_1_1"><i><font color="Green" title="E38">A13</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/>
<b>hence </b><a NAME="E5:53_3_1_1"/>
( <font color="Maroon" title="c5">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> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="txt" href="necklace.html#E5:53"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="necklace.html#E3:53_3_1_1"><i><font color="Green" title="E39">A14</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:53_3_1"/>
( <font color="Maroon" title="c5">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> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> implies ( <font color="Maroon" title="c6">y</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> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:53_3_1_2"/><i><font color="Green" title="E37">A15</font></i>: 
<font color="Maroon" title="c5">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> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E2:53_3_1_2"/><i><font color="Green" title="E38">A16</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 ;<br/>

<a NAME="E3:53_3_1_2"/>
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> 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">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a></i>;<br/>
<a NAME="E4:53_3_1_2"/><b>then </b><i><font color="Green" title="E39">A17</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E1:53_3_1_2"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:53_3_1_2"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E2:53_3_1_2"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/>
<b>hence </b><a NAME="E6:53_3_1_2"/>
( <font color="Maroon" title="c6">y</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> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="necklace.html#E6:53"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="necklace.html#E4:53_3_1_2"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</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="E3:53_3_1"/>
( <font color="Maroon" title="c5">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> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> iff ( <font color="Maroon" title="c6">y</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> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> ) )
 <b>by </b><i><a class="txt" href="necklace.html#E6:53"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="necklace.html#E1:53_3_1"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_3"/>
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="necklace.html#E5:53"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="necklace.html#E6:53"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">h</font> = <font color="Maroon" title="c4">g</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> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 1</span>)</span> <b>by </b><i><a class="txt" href="necklace.html#E3:53"><i><font color="Green" title="E32">A3</font></i></a></i>;<br/>
<a NAME="E10:53"/>
<font color="Maroon" title="c5">h</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="53_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">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> 1</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="E1:53_4"/><i><font color="Green" title="E36">A18</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font>
 ;<br/>

<a NAME="E2:53_4"/><i><font color="Green" title="E37">A19</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> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="necklace.html#E3:53"><i><font color="Green" title="E32">A3</font></i></a>, <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="c8">i</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">j</font> = <font color="Maroon" title="c7">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="E4:53_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E1:53_4"><i><font color="Green" title="E36">A18</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="E5:53_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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> 1</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> 1</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="E6:53_4"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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> 1</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="E7:53_4"/><b>then </b><i><font color="Green" title="E38">A20</font></i>: 
(  not <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> 1</span>)</span> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">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="E8:53_4"/><i><font color="Green" title="E39">A21</font></i>: 
( <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> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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> 1</span>)</span></span>)</span> )
 
;<br/>
<a NAME="E9:53_4"/><i><font color="Green" title="E40">A22</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E2:53_4"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:53_4"/><i><font color="Green" title="E41">A23</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E2:53_4"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:53_4"/><i><font color="Green" title="E42">A24</font></i>: 
( <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">j</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_4_1_1"/><i><font color="Green" title="E43">A25</font></i>: 
( <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> or <font color="Maroon" title="c9">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_4_1_1_1"/>
( <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> or <font color="Maroon" title="c9">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> )
 </span><b>by </b><i><a class="txt" href="necklace.html#E1:53_4_1_1"><i><font color="Green" title="E43">A25</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_4_1_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_4_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="necklace.html#E2:53_4"><i><font color="Green" title="E37">A19</font></i></a>, <a class="txt" href="necklace.html#E9:53_4"><i><font color="Green" title="E40">A22</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_4_1_1_1_2"/>
<font color="Maroon" title="c9">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_4_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="necklace.html#E2:53_4"><i><font color="Green" title="E37">A19</font></i></a>, <a class="txt" href="necklace.html#E10:53_4"><i><font color="Green" title="E41">A23</font></i></a>, <a class="ref" href="euler_1.html#T1" title="EULER_1:th.1">EULER_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:53_4_1"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="necklace.html#E7:53_4"><i><font color="Green" title="E38">A20</font></i></a>, <a class="txt" href="necklace.html#E8:53_4"><i><font color="Green" title="E39">A21</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="c10">a</font>, <font color="Maroon" title="c11">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> 1</span>)</span>;<br/>

<b>assume </b><a NAME="E12:53_4"/>
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font> )
 ;<br/>

<a NAME="E13:53_4"/><i><font color="Green" title="E43">A26</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> 1</span>)</span></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="txt" href="necklace.html#E3:53"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a></i>;<br/>
<a NAME="E14:53_4"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="necklace.html#E2:53_4"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:53_4"/>
<font color="Maroon" title="c10">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">b</font>
 <b>by </b><i><a class="txt" href="necklace.html#E11:53_4"><i><font color="Green" title="E42">A24</font></i></a>, <a class="txt" href="necklace.html#E13:53_4"><i><font color="Green" title="E43">A26</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:53"/><b>then </b>
<font color="Maroon" title="c4">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="necklace.html#E7:53"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/>
<b>hence </b><a NAME="E12:53"/>
 <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 1, <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> 1</span>)</span> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/>


</div>
