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<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E2:48_1_1"/><i><font color="Green" title="E24">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <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>and </b><br/><a NAME="E3:48_1_1"/><i><font color="Green" title="E25">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">M</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">A</font>)
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c5">f</font>
;<br/>

<b>thus </b><a NAME="E4:48_1_1"/>
<font color="Maroon" title="c5">f</font> is    <a href="ordinal1.html#M1" title="ORDINAL1:mode.1">T-Sequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</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>,<a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a><br/>

<b>assume </b><a NAME="E1:48_1_1_1"/>
<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="c5">f</font>
 ;<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="E3:48_1_1_1"/><i><font color="Green" title="E26">A3</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <b>and </b><br/><a NAME="E4:48_1_1_1"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:48_1_1_1"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ami_1.html#D4" title="AMI_1:def.4">AMI_1:def 4</a></i>;<br/>
<a NAME="E6:48_1_1_1"/>
<font color="Maroon" title="c7">x</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:48_1_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>hence </b><a NAME="E7:48_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="amistd_3.html#E2:48_1_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="amistd_3.html#E3:48_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="amistd_3.html#E3:48_1_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="amistd_3.html#E4:48_1_1_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="amistd_3.html#E5:48_1_1_1"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:48_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <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="txt" href="amistd_3.html#E2:48_1_1"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>

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

<b>assume </b><a NAME="E6:48_1_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">M</font>
 ;<br/>

<a NAME="E7:48_1_1"/><b>then </b>
<font color="Maroon" title="c6">m</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>hence </b><a NAME="E8:48_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</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="c6">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:48_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="amistd_3.html#E6:48_1_1"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/>


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