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<b>let </b><font color="Maroon">z</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">N</font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2">void</a> <a href="ami_1.html#V5">IC-Ins-separated</a> <a href="ami_1.html#V8">definite</a> <a href="amistd_1.html#V4" target="_self">standard</a>  <a href="ami_1.html#L1">AMI-Struct</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">P</font> be  <a href="ami_3.html#V1">programmed</a> <a href="amistd_1.html#V9" target="_self">lower</a>  <a href="ami_1.html#M1">FinPartState</a> of <font color="Maroon">T</font>;<br/>







<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>  <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">T</font> =  <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">T</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>;<br/>
<b>set </b><font color="Maroon">A1</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span> ;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>now </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"/><i><font color="Green">E37</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/><a NAME="E2:83_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="ami_3.html#D13">AMI_3:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">n</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="E4:83_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">T</font>,<font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</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>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>thus </b><a NAME="E6:83_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">n</font>)
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E2:83"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">k1</font>, <font color="Olive">k2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k2</font>) holds <br/><font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Olive">k2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:83"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>,<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>) <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 
<b>from </b><i><a class="ref" href="funct_7.html#S3">FUNCT_7:sch 3</a>(, );<br/></i>
<a NAME="E4:83"/><i><font color="Green">E64</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:83_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span> 
 ;<br/><b>assume </b><a NAME="E2:83_2"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E3:83_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">T</font>,<font color="Maroon">b</font> <a href="amistd_1.html#R1" target="_self">&lt;=</a>  <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">T</font>,<font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E4:83_2"/>
 ex <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">T</font>,<font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><a NAME="E5:83_2"/><b>then </b>
 <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">T</font>,<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:83_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span> 
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span>  as   <a href="card_1.html#NM1">Cardinal</a> <b>by </b><i>, <a class="ref" href="funct_7.html#T22">FUNCT_7:22</a></i>;<br/>
<a NAME="E7:83"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">z</font> is    <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/>
<a NAME="E8:83"/><b>then </b><i><font color="Green">E69</font></i>: 
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>
<a NAME="E9:83"/><i><font color="Green">E70</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:83_3"/>
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:83_3"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_1.html#T73">CARD_1:73</a></i>;<br/><a NAME="E3:83_3"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>;<br/><a NAME="E4:83_3"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/><a NAME="E5:83_3"/><b>then </b>
 ex <font color="Olive">d</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">T</font>,<font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:83_3"/>
 <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">T</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E11:83"/>
 <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">T</font>,<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E12:83"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:83"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<a NAME="E14:83"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>;<br/>
<b>hence </b><a NAME="E15:83"/>
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="card_1.html#T73">CARD_1:73</a></i>;<br/>


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