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<b>set </b><font color="Maroon" title="c4">f</font> =  <a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>set </b><font color="Maroon" title="c5">X</font> =  <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>set </b><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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span>;<br/>
<b>let </b><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_1_1"/><i><font color="Green" title="E25">A1</font></i>: 
<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#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <b>and </b><br/><a NAME="E2:50_1_1"/><i><font color="Green" title="E26">A2</font></i>: 
<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#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c3">M</font></span>)</span>
 <b>and </b><br/><a NAME="E3:50_1_1"/><i><font color="Green" title="E27">A3</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c3">M</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#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font>
 ;<br/>

<a NAME="E4:50_1_1"/><i><font color="Green" title="E28">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c3">M</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>
 
<b>by </b><i><a class="ref" href="amistd_3.html#D4" target="_self" title="AMISTD_3:def.4">Def4</a></i>;<br/>
<a NAME="E5:50_1_1"/><b>then </b><i><font color="Green" title="E29">A5</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c3">M</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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</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>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:50_1_1"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="amistd_3.html#D4" target="_self" title="AMISTD_3:def.4">Def4</a></i>;<br/>
<a NAME="E6:50_1_1"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" title="AMISTD_3:func.4">LocSeq</a> <font color="Maroon" title="c3">M</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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</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>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E2:50_1_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="amistd_3.html#E4:50_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="amistd_3.html#D4" target="_self" title="AMISTD_3:def.4">Def4</a></i>;<br/>
<a NAME="E7:50_1_1"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
<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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</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>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:50_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="amistd_3.html#E5:50_1_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="amistd_1.html#T25" title="AMISTD_1:th.25">AMISTD_1:25</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">phi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E9:50_1_1"/><i><font color="Green" title="E31">A7</font></i>: 
<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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E10:50_1_1"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c9">phi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 <b>and </b><br/><a NAME="E11:50_1_1"/><i><font color="Green" title="E33">A9</font></i>: 
 <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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E12:50_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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="card_5.html#T14" title="CARD_5:th.14">CARD_5:14</a></i>;<br/>
<a NAME="E13:50_1_1"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c9">phi</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="amistd_3.html#E10:50_1_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="card_5.html#T20" title="CARD_5:th.20">CARD_5:20</a></i>;<br/>
<a NAME="E14:50_1_1"/>
 <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#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T28" target="_self" title="AMISTD_3:th.28">Th28</a></i>;<br/>
<b>hence </b><a NAME="E15:50_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>
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:50_1_1"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="amistd_3.html#E2:50_1_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="amistd_3.html#E4:50_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="amistd_3.html#E7:50_1_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="amistd_3.html#E9:50_1_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="amistd_3.html#E11:50_1_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="amistd_3.html#E13:50_1_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


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