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<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">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2" target="_self">void</a> <a href="ami_1.html#V5" target="_self">IC-Ins-separated</a> <a href="ami_1.html#V8" target="_self">definite</a>  <a href="ami_1.html#L1" target="_self">AMI-Struct</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">la</font> be   <a href="ami_1.html#NM1">Object</a> of <font color="Maroon">S</font>, <font color="Maroon">lb</font> be   <a href="ami_1.html#NM1">Object</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:86"/><i><font color="Green">E28</font></i>: 
(  <a href="ami_1.html#K3" target="_self">ObjectKind</a> <font color="Maroon">la</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="ami_1.html#K3" target="_self">ObjectKind</a> <font color="Maroon">lb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/>



<b>set </b><font color="Maroon">p</font> = <font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<a NAME="E2:86"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<a NAME="E3:86"/><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="ami_1.html#U5" target="_self">Object-Kind</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_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="E1:86_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:86_1_1_1"/>
( ( <font color="Maroon">la</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">lb</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">la</font> ) or ( <font color="Maroon">la</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">lb</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lb</font> ) or ( <font color="Maroon">la</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lb</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">la</font> ) )
 </span><b>by </b><i>, <a class="ref" href="ami_1.html#T11" target="_self">Th11</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_1"><b>suppose </b></a><a NAME="E1:86_1_1_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">la</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">lb</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">la</font> )
 ;<br/><div class="add"><a NAME="E2:86_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><a NAME="E3:86_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K3" target="_self">ObjectKind</a> <font color="Maroon">la</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:86_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5" target="_self">Object-Kind</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_2"><b>suppose </b></a><a NAME="E1:86_1_1_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">la</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">lb</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lb</font> )
 ;<br/><div class="add"><a NAME="E2:86_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T66">FUNCT_4:66</a></i>;<br/><a NAME="E3:86_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K3" target="_self">ObjectKind</a> <font color="Maroon">lb</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:86_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5" target="_self">Object-Kind</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1_3"><b>suppose </b></a><a NAME="E1:86_1_1_1_3"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">la</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lb</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">la</font> )
 ;<br/><div class="add"><a NAME="E2:86_1_1_1_3"/><b>then </b>
<font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">la</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T44">CQC_LANG:44</a></i>;<br/><a NAME="E3:86_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/><a NAME="E4:86_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K3" target="_self">ObjectKind</a> <font color="Maroon">lb</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:86_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5" target="_self">Object-Kind</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:86_1"/>
<span class="p1">(<span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U5" target="_self">Object-Kind</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ami_1.html#K7" target="_self">sproduct</a> the <a href="ami_1.html#U5" target="_self">Object-Kind</a> of <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="ami_1.html#T10" target="_self">Th10</a>, <a class="ref" href="ami_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E6:86"/>
<font color="Maroon">p</font> is    <a href="ami_1.html#M1" target="_self">FinPartState</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>proof </b></a><div class="add">

<a NAME="E1:86_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">la</font>,<font color="Maroon">lb</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<b>hence </b><a NAME="E2:86_2"/>
<font color="Maroon">p</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T29">FINSET_1:29</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ami_1.html#D24" target="_self">AMI_1:def 24</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:86"/>
<font color="Maroon">la</font>,<font color="Maroon">lb</font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is    <a href="ami_1.html#M1" target="_self">FinPartState</a> of <font color="Maroon">S</font>
 ;<br/>


</div>
