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<b>let </b><font color="Maroon" title="c1">N</font> be  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="ami_1.html#V2" title="AMI_1:attr.2">void</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c1">N</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c5">la</font>, <font color="Maroon" title="c6">lb</font> be   <a href="ami_1.html#NM1" title="AMI_1:NM.1">Object</a> of <font color="Maroon" title="c2">S</font>;<br/>







<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E29">A1</font></i>: 
(  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Maroon" title="c5">la</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Maroon" title="c6">lb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c7">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font>;<br/><b>let </b><font color="Maroon" title="c8">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">B</font>;<br/>



<b>set </b><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>;<br/>
<a NAME="E2:59"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>;<br/>
<a NAME="E3:59"/><i><font color="Green" title="E31">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:59_1"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1_1"/>
( ( <font color="Maroon" title="c5">la</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">lb</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">la</font> ) or ( <font color="Maroon" title="c5">la</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">lb</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">lb</font> ) or ( <font color="Maroon" title="c5">la</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">lb</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">la</font> ) )
 </span><b>by </b><i><a class="txt" href="ami_1.html#E2:59"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="ami_1.html#E1:59_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_1_1_1"/><i><font color="Green" title="E33">A5</font></i>: 
( <font color="Maroon" title="c5">la</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">lb</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">la</font> )
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E1:59_1_1_1_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="ami_1.html#E1:59"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_1_1_2"/><i><font color="Green" title="E33">A6</font></i>: 
( <font color="Maroon" title="c5">la</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">lb</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">lb</font> )
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>;<br/><b>hence </b><a NAME="E3:59_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E1:59_1_1_1_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="ami_1.html#E1:59"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_3"><b>suppose </b></a><a NAME="E1:59_1_1_1_3"/><i><font color="Green" title="E33">A7</font></i>: 
( <font color="Maroon" title="c5">la</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">lb</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">la</font> )
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T44" title="CQC_LANG:th.44">CQC_LANG:44</a></i>;<br/><a NAME="E3:59_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E1:59_1_1_1_3"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>;<br/><b>hence </b><a NAME="E4:59_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E1:59_1_1_1_3"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="ami_1.html#E1:59"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">p</font> = <font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U4" title="AMI_1:sel.4">Object-Kind</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="ami_1.html#E2:59"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="ami_1.html#E3:59"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="card_3.html#D9" title="CARD_3:def.9">CARD_3:def 9</a></i>;<br/>
<a NAME="E6:59"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>;<br/>
<a NAME="E7:59"/><b>then </b>
<font color="Maroon" title="c10">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T29" title="FINSET_1:th.29">FINSET_1:29</a></i>;<br/>
<b>hence </b><a NAME="E8:59"/>
<font color="Maroon" title="c5">la</font>,<font color="Maroon" title="c6">lb</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> is   <a href="ami_1.html#NM5" title="AMI_1:NM.5">FinPartState</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E81"><i><font color="Green" title="E24">Lm1</font></i></a></i>;<br/>


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