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

<b>set </b><font color="Maroon">f1</font> =  <a href="twoscomp.html#K3">and2a</a> ;<br/>
<b>set </b><font color="Maroon">f2</font> =  <a href="gfacirc1.html#K3" target="_self">and2c</a> ;<br/>
<b>set </b><font color="Maroon">f3</font> =  <a href="twoscomp.html#K4">and2b</a> ;<br/>
<b>set </b><font color="Maroon">f4</font> =  <a href="twoscomp.html#K29">nor3</a> ;<br/>
<b>let </b><font color="Maroon">x</font> be  non <a href="facirc_1.html#V1">pair</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be  non <a href="facirc_1.html#V1">pair</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be  non <a href="facirc_1.html#V1">pair</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">xy</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span>;<br/>
<b>set </b><font color="Maroon">yz</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span>;<br/>
<b>set </b><font color="Maroon">zx</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span><a href="tarski.html#K4">]</a></span>;<br/>
<b>set </b><font color="Maroon">Cxy</font> =  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> ;<br/>
<b>set </b><font color="Maroon">Cyz</font> =  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> ;<br/>
<b>set </b><font color="Maroon">Czx</font> =  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> ;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="gfacirc1.html#K31" target="_self">GFA2CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">MI</font> =  <a href="gfacirc1.html#K29" target="_self">GFA2CarryIStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K7">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span><a href="tarski.html#K4">]</a></span></span><a href="facirc_1.html#K7">*&gt;</a></span>,<a href="twoscomp.html#K29">nor3</a> ;<br/>
<a NAME="E1:117"/><i><font color="Green">E27</font></i>: 
( not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span>)</span> is <a href="facirc_1.html#V2">with_pair</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span>)</span> is <a href="facirc_1.html#V2">with_pair</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span>)</span> is <a href="facirc_1.html#V2">with_pair</a> )
 
<b>by </b><i><a class="ref" href="facirc_1.html#T41">FACIRC_1:41</a></i>;<br/>
<a NAME="E2:117"/><b>then </b><i><font color="Green">E32</font></i>: 
not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span>)</span> <a href="circcomb.html#K3">+*</a> <span class="p2">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span>)</span></span>)</span> is <a href="facirc_1.html#V2">with_pair</a>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T9">FACIRC_1:9</a></i>;<br/>
<a NAME="E3:117"/><b>then </b><i><font color="Green">E33</font></i>: 
not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K29" target="_self">GFA2CarryIStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> is <a href="facirc_1.html#V2">with_pair</a>
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E18"><i><font color="Green">Lemma70</font></i></a>, <a class="ref" href="facirc_1.html#T9">FACIRC_1:9</a></i>;<br/>
<a NAME="E4:117"/><i><font color="Green">E35</font></i>: 
(  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="facirc_1.html#T40">FACIRC_1:40</a></i>;<br/>
<a NAME="E5:117"/>
 <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K7">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span><a href="tarski.html#K4">]</a></span></span><a href="facirc_1.html#K7">*&gt;</a></span>,<a href="twoscomp.html#K29">nor3</a> </span>)</span> is   <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T38">FACIRC_1:38</a></i>;<br/>
<a NAME="E6:117"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K31" target="_self">GFA2CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <span class="p2">(<span class="default"><a href="gfacirc1.html#K29" target="_self">GFA2CarryIStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <span class="p3">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p4"><a href="facirc_1.html#K7">&lt;*</a><span class="default"><span class="p5"><a href="tarski.html#K4">[</a><span class="default"><span class="p0"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p5"><a href="tarski.html#K4">[</a><span class="default"><span class="p0"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p5"><a href="tarski.html#K4">[</a><span class="default"><span class="p0"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span><a href="tarski.html#K4">]</a></span></span><a href="facirc_1.html#K7">*&gt;</a></span>,<a href="twoscomp.html#K29">nor3</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <span class="p3">(<span class="default"><a href="gfacirc1.html#K29" target="_self">GFA2CarryIStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="facirc_1.html#T6">FACIRC_1:6</a></i>;<br/>
<a NAME="E7:117"/><i><font color="Green">E55</font></i>: 
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K7">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span><a href="tarski.html#K4">]</a></span></span><a href="facirc_1.html#K7">*&gt;</a></span>,<a href="twoscomp.html#K29">nor3</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T42">FACIRC_1:42</a></i>;<br/>
<a NAME="E8:117"/><i><font color="Green">E56</font></i>: 
(  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span>)</span> <a href="hidden.html#R1">=</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"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span>)</span> <a href="hidden.html#R1">=</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"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span>)</span> <a href="hidden.html#R1">=</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"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </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#T49">CIRCCOMB:49</a></i>;<br/>
<a NAME="E9:117"/>
(  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a>  <a href="circcomb.html#R1">tolerates</a>  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a>  &amp;  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a>  <a href="circcomb.html#R1">tolerates</a>  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a>  &amp;  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a>  <a href="circcomb.html#R1">tolerates</a>  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a>  )
 
<b>by </b><i><a class="ref" href="circcomb.html#T55">CIRCCOMB:55</a></i>;<br/>
<a NAME="E10:117"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span>)</span> <a href="circcomb.html#K3">+*</a> <span class="p2">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</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"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</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"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </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#T15">CIRCCOMB:15</a></i>;<br/>
<a NAME="E11:117"/>
<span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span>)</span> <a href="circcomb.html#K3">+*</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span>)</span> <a href="circcomb.html#R1">tolerates</a>  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> 
 
<b>by </b><i><a class="ref" href="circcomb.html#T55">CIRCCOMB:55</a></i>;<br/>
<b>then </b> <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K29" target="_self">GFA2CarryIStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</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"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </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#T15">CIRCCOMB:15</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</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"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T43">ENUMSET1:43</a></i>
;<br/>
<a NAME="E13:117"/><b>then </b>
<span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <span class="p2">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K7">&lt;*</a><span class="default"><span class="p4"><a href="tarski.html#K4">[</a><span class="default"><span class="p5"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p4"><a href="tarski.html#K4">[</a><span class="default"><span class="p5"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p4"><a href="tarski.html#K4">[</a><span class="default"><span class="p5"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span><a href="tarski.html#K4">]</a></span></span><a href="facirc_1.html#K7">*&gt;</a></span>,<a href="twoscomp.html#K29">nor3</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <span class="p2">(<span class="default"><a href="gfacirc1.html#K29" target="_self">GFA2CarryIStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b> <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K31" target="_self">GFA2CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span>)</span> <a href="circcomb.html#K3">+*</a> <span class="p3">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <span class="p2">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gfacirc1.html#E18"><i><font color="Green">Lemma70</font></i></a>, , , , , <a class="ref" href="facirc_1.html#T7">FACIRC_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <span class="p3">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3">and2a</a> </span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <span class="p3">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p4"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K3" target="_self">and2c</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <span class="p2">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K4">and2b</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gfacirc1.html#E18"><i><font color="Green">Lemma70</font></i></a>, , <a class="ref" href="facirc_1.html#T7">FACIRC_1:7</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="txt" href="gfacirc1.html#E19"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="enumset1.html#T45">ENUMSET1:45</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T110">ENUMSET1:110</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T71">ENUMSET1:71</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T99">ENUMSET1:99</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T100">ENUMSET1:100</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">z</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="enumset1.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T44">ENUMSET1:44</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T71">ENUMSET1:71</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T100">ENUMSET1:100</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T44">ENUMSET1:44</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T71">ENUMSET1:71</a></i>
;<br/><br/>


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