<?xml version="1.0"?>
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<span class="kw">set </span><font color="Maroon" title="c4">X</font> =  <a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">C</font> =  <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span> or  not <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span> or  not <span class="p1">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font> or <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_1_1"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E2:37_1_1"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font></span><br/>

<a NAME="E3:37_1_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="amistd_3.html#D3" target="_self" title="AMISTD_3:def.3">Def4</a></span>;<br/>
<a NAME="E4:37_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p5">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p0">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocSeq</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p5">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p0">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="amistd_3.html#D3" target="_self" title="AMISTD_3:def.3">Def4</a></span>;<br/>
<a NAME="E5:37_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<span class="p1">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p5">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p5">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p3">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="amistd_1.html#T25" title="AMISTD_1:th.25">AMISTD_1:25</a></span>;<br/>
<a NAME="E6:37_1_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="amistd_3.html#T28" target="_self" title="AMISTD_3:th.28">Th28</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">phi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><span class="kw"> such that </span><br/><a NAME="E8:37_1_1"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c9">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord1.html#K3" title="WELLORD1:func.3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:37_1_1"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c9">phi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 <span class="kw">and </span><br/><a NAME="E10:37_1_1"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wellord2.html#K2" title="WELLORD2:func.2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1" title="WELLORD2:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:37_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_3.html#K2" title="AMISTD_3:func.2">LocNums</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T14" title="CARD_5:th.14">CARD_5:14</a></span>;<br/>
<a NAME="E12:37_1_1"/>
<font color="Maroon" title="c9">phi</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_1_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="card_5.html#T20" title="CARD_5:th.20">CARD_5:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:37_1_1"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E5:37_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E8:37_1_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E10:37_1_1"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E6:37_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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