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<b>let </b><font color="Maroon" title="c5">IT1</font>, <font color="Maroon" title="c6">IT2</font> be   <a href="ami_1.html#NM7" title="AMI_1:NM.7">FinPartState</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:75_1_2"/><i><font color="Green" title="E39">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font>
 <b>and </b><br/><a NAME="E2:75_1_2"/><i><font color="Green" title="E40">A10</font></i>: 
for <font color="Olive" title="b1">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">m</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="c3">p</font> holds <br/><font color="Maroon" title="c5">IT1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">k</font>
 <b>and </b><br/><a NAME="E3:75_1_2"/><i><font color="Green" title="E41">A11</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">IT2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font>
 <b>and </b><br/><a NAME="E4:75_1_2"/><i><font color="Green" title="E42">A12</font></i>: 
for <font color="Olive" title="b1">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">m</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="c3">p</font> holds <br/><font color="Maroon" title="c6">IT2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">k</font>
 ;<br/>

<a NAME="E5:75_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c3">p</font> holds <br/><font color="Maroon" title="c5">IT1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:75_1_2_1"/><i><font color="Green" title="E43">A13</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="c3">p</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">l</font> = <font color="Maroon" title="c7">x</font> as    <a href="ami_1.html#M3" title="AMI_1:mode.3">Instruction-Location</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="amistd_2.html#E1:75"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="ami_1.html#D4" title="AMI_1:def.4">AMI_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">m</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:75_1_2_1"/><i><font color="Green" title="E44">A14</font></i>: 
<font color="Maroon" title="c8">l</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>,<font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="ref" href="amistd_1.html#T26" title="AMISTD_1:th.26">AMISTD_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">m</font> = <font color="Maroon" title="c9">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c5">IT1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="amistd_2.html#K7" title="AMISTD_2:func.7">IncAddr</a> <span class="p1">(<span class="default"><a href="ami_1.html#K31" title="AMI_1:func.31">pi</a> <font color="Maroon" title="c3">p</font>,<span class="p2">(<span class="default"><a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c10">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="txt" href="amistd_2.html#E2:75_1_2"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="amistd_2.html#E1:75_1_2_1"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="amistd_2.html#E4:75_1_2_1"><i><font color="Green" title="E44">A14</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">IT2</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="txt" href="amistd_2.html#E4:75_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="amistd_2.html#E1:75_1_2_1"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="amistd_2.html#E4:75_1_2_1"><i><font color="Green" title="E44">A14</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:75_1_2"/>
<font color="Maroon" title="c5">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E1:75_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="amistd_2.html#E3:75_1_2"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
