<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V5">gate`2=den</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">S</font>;<br/>



<b>assume </b><a NAME="E1:69"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> is <a href="circcomb.html#V4">gate`2=den</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> or ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> ) );<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="relat_1.html#NM1">Relation</a><b> such that </b><br/><a NAME="E3:69"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] ) )
 <b>from </b><i><a class="ref" href="relat_1.html#S1">RELAT_1:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> implies  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">s</font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font><br/> for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E4:69"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">R</font> is <a href="rewrite1.html#V1">co-well_founded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D16">REWRITE1:def 16</a><br/>

<b>assume </b><a NAME="E1:69_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:69_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
;<br/>
<b>consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:69_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> iff  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="tarski.html#S1">TARSKI:sch 1</a>(<a class="txt" href="circcmb3.html#E1:69_1"><i><font color="Green">E38</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/>
<a NAME="E6:69_1"/>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E4:69"><i><font color="Green">E37</font></i></a>, <a class="ref" href="circcmb3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E7:69_1"/><b>then </b>
 <a href="ordinal2.html#K2">inf</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T25">ORDINAL2:25</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:69_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  <a href="ordinal2.html#K2">inf</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="circcmb3.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E10:69_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="circcmb3.html#D2" target="_self">Def2</a></i>;<br/>

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

<b>assume </b><a NAME="E11:69_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<a NAME="E12:69_1"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:69_1"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a> 
 <b>by </b><i><a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<a NAME="E15:69_1"/><i><font color="Green">E65</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="classes1.html#T76">CLASSES1:76</a></i>;<br/>
<a NAME="E16:69_1"/><i><font color="Green">E66</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E17:69_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default">the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E1:69"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="circcomb.html#D10">CIRCCOMB:def 10</a></i>;<br/>
<a NAME="E18:69_1"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="classes1.html#K6">the_rank_of</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:69_1"/>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="circcmb3.html#D1" target="_self">Def1</a>, <a class="ref" href="circcmb3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E20:69_1"/><b>then </b>
 <a href="ordinal2.html#K2">inf</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T22">ORDINAL2:22</a></i>;<br/>
<b>hence </b><a NAME="E21:69_1"/>
contradiction
 <b>by </b><i><a class="ref" href="circcmb3.html#D2" target="_self">Def2</a>, , <a class="txt" href="circcmb3.html#E2:69_1"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:69"/><i><font color="Green">E116</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:69_2"/><i><font color="Green">E117</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>thus </b><a NAME="E4:69_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:69"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:69_3"/><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">o</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E4:69_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:69"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">R</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E4:69"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E8:69"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
;<br/>
<a NAME="E9:69"/><i><font color="Green">E120</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_4"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:69_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">o</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:69_4"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:69"/><b>then </b><i><font color="Green">E123</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E11:69"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st ( for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:69_5"/><i><font color="Green">E134</font></i>: 
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> );<br/>
<b>consider </b><font color="Maroon">RS</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:69_5"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RS</font> iff ( <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<b>assume </b><a NAME="E4:69_5"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<a NAME="E5:69_5"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">RS</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_5_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RS</font>
 ;<br/>

<b>hence </b><a NAME="E2:69_5_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="circcmb3.html#D2" target="_self">Def2</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:69_5"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">RS</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; ( for <font color="Olive">s</font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font><br/> for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E7:69_5"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RS</font> holds <br/> ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:69_5_2"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RS</font>
 ;<br/>

<a NAME="E2:69_5_2"/><b>then </b>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:69_5_2"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">s</font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font><br/> for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="circcmb3.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E4:69_5_2"/>
 ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">b</font>,<font color="Olive">z</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:69_5"/><i><font color="Green">E142</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">RS</font>
 <b>and </b><br/><a NAME="E10:69_5"/><i><font color="Green">E143</font></i>: 
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> <font color="Maroon">RS</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_5_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_5_3"/>
( <font color="Maroon">RS</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">RS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_5_3_1"><b>suppose </b></a><a NAME="E1:69_5_3_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">RS</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:69_5_3_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_5_3_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">l</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:69_5_3_1_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:69_5_3_1_1"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">s</font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font><br/> for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> ) )
 <b>by </b><i><a class="ref" href="circcmb3.html#D5" target="_self">Def5</a>, , <a class="ref" href="circcmb3.html#T2" target="_self">Th2</a></i>;<br/>
<b>thus </b><a NAME="E6:69_5_3_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="circcmb3.html#T2" target="_self">Th2</a>, <a class="ref" href="circcmb3.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">C</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="circcmb3.html#D5" target="_self">Def5</a>, , <a class="ref" href="finset_1.html#T26">FINSET_1:26</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><b>reconsider </b><font color="Maroon">mC</font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">C</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">mC</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>let </b><font color="Maroon">s</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:69_5_3_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:69_5_3_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</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>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">RS</font>;<br/><a NAME="E10:69_5_3_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as   <a href="circcomb.html#NM1">Gate</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><i><font color="Green">E151</font></i>:  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">a1</font> = 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="circcomb.html#T52">CIRCCOMB:52</a></i>
;<br/><a NAME="E13:69_5_3_1"/>
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#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a1</font></span>)</span> holds <br/> <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">mC</font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5_3_1_3"><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="E1:69_5_3_1_3"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a1</font></span>)</span>
 ;<br/>

<a NAME="E2:69_5_3_1_3"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_5.html#T21">FUNCT_5:21</a></i>;<br/>
<a NAME="E3:69_5_3_1_3"/>
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="circcomb.html#D8">CIRCCOMB:def 8</a></i>;<br/>
<a NAME="E4:69_5_3_1_3"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E5:69_5_3_1_3"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="circcmb3.html#T5" target="_self">Th5</a>, <a class="ref" href="circcomb.html#T50">CIRCCOMB:50</a></i>;<br/>
<a NAME="E6:69_5_3_1_3"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 
;<br/>
<a NAME="E7:69_5_3_1_3"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a1</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E8:69_5_3_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="circcmb3.html#T5" target="_self">Th5</a>, <a class="ref" href="circcmb3.html#T8" target="_self">Th8</a>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E9:69_5_3_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:69_5_3_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E11:69_5_3_1_3"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RS</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:69_5_3_1_3"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> &amp; ( for <font color="Olive">s</font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font><br/> for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font> holds <br/><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E14:69_5_3_1_3"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="circcmb3.html#D5" target="_self">Def5</a>, , , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E15:69_5_3_1_3"/><b>then </b><i><font color="Green">E158</font></i>: 
 <a href="pre_circ.html#K1">max</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5_3_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:69_5_3_1_3_1"/>
<font color="Maroon">mC</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E2:69_5_3_1_3_1"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">mC</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">mC</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><font color="Maroon">mC</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="facirc_1.html#T13">FACIRC_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">mC</font></span>)</span> <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="E17:69_5_3_1_3"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">mC</font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:69_5_3_1"/><b>then </b>
 <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">mC</font></span>)</span> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="circcmb3.html#T4" target="_self">Th4</a>, <a class="ref" href="facirc_1.html#T19">FACIRC_1:19</a></i>;<br/><a NAME="E15:69_5_3_1"/><b>then </b>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="facirc_1.html#T12">FACIRC_1:12</a></i>;<br/><a NAME="E16:69_5_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a></i>;<br/><b>hence </b><a NAME="E17:69_5_3_1"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="circcmb3.html#T2" target="_self">Th2</a>, <a class="ref" href="facirc_1.html#T13">FACIRC_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_5_3_2"><b>suppose </b></a><a NAME="E1:69_5_3_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">RS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">n</font> = 1;<br/><b>let </b><font color="Maroon">s</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E2:69_5_3_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:69_5_3_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</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/><a NAME="E6:69_5_3_2"/><i><font color="Green">E162</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><div><i><font color="Green">E163</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_5_3_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:69_5_3_2_1"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">s</font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="facirc_1.html#T18">FACIRC_1:18</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="circcmb3.html#T8" target="_self">Th8</a>, <a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5_3_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:69_5_3_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:69_5_3_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font>
 ;<br/><a NAME="E3:69_5_3_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="circcomb.html#D7">CIRCCOMB:def 7</a></i>;<br/><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> as   <a href="circcomb.html#NM1">Gate</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><i><font color="Green">E165</font></i>:  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">a1</font> = 
the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D7">MSUALG_1:def 7</a></i>
<br/>.= 
<font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="circcomb.html#T52">CIRCCOMB:52</a></i>
;<br/><a NAME="E6:69_5_3_2_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#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a1</font></span>)</span> holds <br/><font color="Maroon">s</font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_5_3_2_2_2"><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="E1:69_5_3_2_2_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a1</font></span>)</span>
 ;<br/>

<a NAME="E2:69_5_3_2_2_2"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_5.html#T21">FUNCT_5:21</a></i>;<br/>
<a NAME="E3:69_5_3_2_2_2"/>
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="circcomb.html#D8">CIRCCOMB:def 8</a></i>;<br/>
<a NAME="E4:69_5_3_2_2_2"/><b>then </b><i><font color="Green">E168</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D6">MSUALG_1:def 6</a></i>;<br/>
<a NAME="E5:69_5_3_2_2_2"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">a1</font>
 
<b>by </b><i>, <a class="ref" href="circcomb.html#T50">CIRCCOMB:50</a></i>;<br/>
<a NAME="E6:69_5_3_2_2_2"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 
;<br/>
<a NAME="E7:69_5_3_2_2_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <font color="Maroon">a1</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E8:69_5_3_2_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E9:69_5_3_2_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:69_5_3_2_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>hence </b><a NAME="E11:69_5_3_2_2_2"/>
<font color="Maroon">s</font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E7:69_5_3_2_2"/><b>then </b>
 <a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font> <a href="facirc_1.html#R1">is_stable_at</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="facirc_1.html#T19">FACIRC_1:19</a></i>;<br/><a NAME="E8:69_5_3_2_2"/><b>then </b>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">a1</font>
 <b>by </b><i><a class="ref" href="circcmb3.html#T8" target="_self">Th8</a>, <a class="ref" href="facirc_1.html#T14">FACIRC_1:14</a></i>;<br/><a NAME="E9:69_5_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a></i>;<br/><b>hence </b><a NAME="E10:69_5_3_2_2"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="circcmb3.html#T2" target="_self">Th2</a>, <a class="ref" href="facirc_1.html#T13">FACIRC_1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:69_5_3_2"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="circcmb3.html#T4" target="_self">Th4</a>, <a class="ref" href="circcmb3.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:69"/><i><font color="Green">E170</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>]
 
<b>from </b><i><a class="ref" href="rewrite1.html#S2">REWRITE1:sch 2</a>(<a class="txt" href="circcmb3.html#E3:69"><i><font color="Green">E36</font></i></a>, <a class="ref" href="circcmb3.html#D4" target="_self">Def4</a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; ( for <font color="Olive">s</font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font><br/> for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E13:69"/><i><font color="Green">E171</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> holds <br/> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:69_6"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:69_6"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">s</font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font><br/> for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:69_6"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">a</font>,<font color="Olive">b</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E15:69"/><i><font color="Green">E172</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E16:69"/><i><font color="Green">E173</font></i>: 
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#K3">field</a> <font color="Maroon">R</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<a NAME="E17:69"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_7"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">l</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:69_7"/><i><font color="Green">E174</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:69_7"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">s</font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font><br/> for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> ) )
 <b>by </b><i>, , </i>;<br/>
<b>thus </b><a NAME="E6:69_7"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">C</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="circcmb3.html#D2" target="_self">Def2</a>, , <a class="ref" href="finset_1.html#T26">FINSET_1:26</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<b>reconsider </b><font color="Maroon">N</font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">C</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>
; <i><font color="Red">:: according to </font></i><a class="ref" href="circcmb3.html#D3" target="_self">CIRCCMB3:def 3</a><br/>

<b>let </b><font color="Maroon">s</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E20:69"/><i><font color="Green">E177</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">N</font></span>)</span></span>)</span> <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="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_8"><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="E1:69_8"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:69_8"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">s</font> being  <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font><br/> for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Olive">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Olive">s</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> ) )
 <b>by </b><i><a class="ref" href="circcmb3.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E4:69_8"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="circcmb3.html#D1" target="_self">Def1</a>, , , <a class="ref" href="circcmb3.html#D5" target="_self">Def5</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:69_8"/><b>then </b><i><font color="Green">E180</font></i>: 
<font color="Maroon">N</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E6:69_8"/><b>then </b><i><font color="Green">E181</font></i>: 
<font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">N</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="circcmb3.html#D5" target="_self">Def5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="circcmb3.html#D5" target="_self">Def5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">N</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="facirc_1.html#T12">FACIRC_1:12</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E22:69"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">N</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circuit2.html#D6">CIRCUIT2:def 6</a><br/>


</div>
