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

<b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>set </b><font color="Maroon">S</font> =  <a href="circcomb.html#K6">1GateCircStr</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>,<font color="Maroon">x</font>;<br/>
<b>thus </b><a NAME="E1:78"/>
 <a href="circcomb.html#K6">1GateCircStr</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>,<font color="Maroon">x</font> is <a href="circcomb.html#V2">gate`1=arity</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="circcomb.html#D8">CIRCCOMB:def 8</a><br/>

<b>assume </b><a NAME="E1:78_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K6">1GateCircStr</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span>
 ;<br/>

<a NAME="E2:78_1"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="circcomb.html#D5">CIRCCOMB:def 5</a></i>;<br/>
<a NAME="E3:78_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:78_1"/>
<font color="Maroon">g</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 <span class="p3">(<span class="default"><a href="circcomb.html#K6">1GateCircStr</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</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#D5">CIRCCOMB:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:78"/>
 <a href="circcomb.html#K6">1GateCircStr</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>,<font color="Maroon">x</font> is <a href="msafree2.html#V2">Circuit-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S'</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree2.html#D6">MSAFREE2:def 6</a><br/>

<b>assume </b><a NAME="E1:78_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S'</font> <a href="hidden.html#R1">=</a>  <a href="circcomb.html#K6">1GateCircStr</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>,<font color="Maroon">x</font>
 ;<br/>

<b>let </b><font color="Maroon">o1</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S'</font>, <font color="Maroon">o2</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S'</font>;<br/>

<a NAME="E2:78_2"/>
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i>, <a class="ref" href="circcomb.html#T44">CIRCCOMB:44</a></i>;<br/>
<b>hence </b><a NAME="E3:78_2"/>
(  not  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o2</font> or <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> )
 ;<br/>


</div><b>end;</b></div>


</div>
