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<b>let </b><font color="Maroon">IT1</font> be    <a href="ami_1.html#M1">FinPartState</a> of <font color="Maroon">S</font>, <font color="Maroon">IT2</font> be    <a href="ami_1.html#M1">FinPartState</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:83_1_2"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">m</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="amistd_2.html#K8" target="_self">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">m</font></span>)</span></span>)</span>,<font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E3:83_1_2"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">IT2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E4:83_1_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">m</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="amistd_2.html#K8" target="_self">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">m</font></span>)</span></span>)</span>,<font color="Maroon">k</font>
 ;<br/>

<a NAME="E5:83_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:83_1_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:83_1_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="amistd_2.html#D1" target="_self">Def1</a>, <a class="ref" href="amistd_1.html#T26">AMISTD_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>thus </b><font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
 <a href="amistd_2.html#K8" target="_self">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_5.html#K5">pi</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Maroon">m</font></span>)</span></span>)</span>,<font color="Maroon">k</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:83_1_2"/>
<font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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