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<b>set </b><font color="Maroon">f</font> =  <a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">M</font>;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span>;<br/>
<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:52_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">M</font></span>)</span>
 <b>and </b><br/><a NAME="E2:52_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">M</font></span>)</span>
 <b>and </b><br/><a NAME="E3:52_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E4:52_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:52_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p4">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p5">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p0">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:52_1_1"/>
<span class="p1">(<span class="default"><a href="amistd_3.html#K4" target="_self">LocSeq</a> <font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p4">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p5">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p0">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:52_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p5">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p4">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p5">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p3">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 
<b>by </b><i>, , <a class="ref" href="amistd_1.html#T25">AMISTD_1:25</a></i>;<br/>
<b>consider </b><font color="Maroon">phi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E9:52_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">phi</font> <a href="hidden.html#R1">=</a>  <a href="wellord1.html#K3">canonical_isomorphism_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="wellord2.html#K2">order_type_of</a> <span class="p3">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p4">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E10:52_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">phi</font> is <a href="ordinal2.html#V2">increasing</a>
 <b>and </b><br/><a NAME="E11:52_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> <a href="hidden.html#R1">=</a>  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E12:52_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font> <a href="hidden.html#R1">=</a>  <a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_5.html#T14">CARD_5:14</a></i>;<br/>
<a NAME="E13:52_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">phi</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="amistd_3.html#T1" target="_self">Th1</a>, <a class="ref" href="card_5.html#T20">CARD_5:20</a></i>;<br/>
<a NAME="E14:52_1_1"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">M</font> <a href="ordinal1.html#R1">c=</a>  <a href="wellord2.html#K2">order_type_of</a> <span class="p1">(<span class="default"><a href="wellord2.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="amistd_3.html#K3" target="_self">LocNums</a> <font color="Maroon">M</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:52_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , , , , <a class="txt" href="amistd_3.html#E1:52_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="amistd_3.html#E2:52_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


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