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

<b>set </b><font color="Maroon" title="c1">f1</font> =  <a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> ;<br/>
<b>set </b><font color="Maroon" title="c2">f2</font> =  <a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> ;<br/>
<b>set </b><font color="Maroon" title="c3">f3</font> =  <a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> ;<br/>
<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c4">x</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/>

<b>set </b><font color="Maroon" title="c7">xy</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c8">yz</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c9">zx</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c4">x</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<b>let </b><font color="Maroon" title="c10">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K8" title="GFACIRC1:func.8">GFA0CarryCirc</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c11">a1</font>, <font color="Maroon" title="c12">a2</font>, <font color="Maroon" title="c13">a3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="circcomb.html#K10" title="CIRCCOMB:func.10">BOOLEAN</a> ;<br/>



<b>assume </b><a NAME="E2:35"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c11">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c12">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c13">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c14">S</font> =  <a href="gfacirc1.html#K7" title="GFACIRC1:func.7">GFA0CarryStr</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x'</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c16">y'</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c17">z'</font> = <font color="Maroon" title="c6">z</font> as   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K7" title="GFACIRC1:func.7">GFA0CarryStr</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <b>by </b><i><a class="ref" href="gfacirc1.html#T19" target="_self" title="GFACIRC1:th.19">Th19</a></i>;<br/>
<a NAME="E4:35"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K7" title="GFACIRC1:func.7">GFA0CarryStr</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K7" title="GFACIRC1:func.7">GFA0CarryStr</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K7" title="GFACIRC1:func.7">GFA0CarryStr</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E1:35"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="gfacirc1.html#T21" target="_self" title="GFACIRC1:th.21">Th21</a></i>;<br/>
<a NAME="E5:35"/><b>then </b><i><font color="Green" title="E25">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="twoscomp.html#K1" title="TWOSCOMP:func.1">.</a> <font color="Maroon" title="c15">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="twoscomp.html#K1" title="TWOSCOMP:func.1">.</a> <font color="Maroon" title="c16">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="twoscomp.html#K1" title="TWOSCOMP:func.1">.</a> <font color="Maroon" title="c17">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> )
 
<b>by </b><i><a class="ref" href="circuit2.html#D5" title="CIRCUIT2:def.5">CIRCUIT2:def 5</a></i>;<br/>
<a NAME="E6:35"/><i><font color="Green" title="E26">A4</font></i>: 
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c10">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T15" title="FACIRC_1:th.15">FACIRC_1:15</a></i>;<br/>
<a NAME="E7:35"/>
( <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c12">a2</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c13">a3</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c4">x</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a3</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c11">a1</font> )
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E2:35"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="gfacirc1.html#T23" target="_self" title="GFACIRC1:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E8:35"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span> <a href="twoscomp.html#K1" title="TWOSCOMP:func.1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K9" title="GFACIRC1:func.9">GFA0CarryOutput</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c12">a2</font></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">a2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c13">a3</font></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a3</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c11">a1</font></span>)</span>
 <b>by </b><i><a class="txt" href="gfacirc1.html#E6:35"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="gfacirc1.html#T24" target="_self" title="GFACIRC1:th.24">Th24</a></i>;<br/>

<b>thus </b><a NAME="E9:35"/>
( <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c12">a2</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a2</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c13">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c6">z</font>,<font color="Maroon" title="c4">x</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a3</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c11">a1</font> )
 <b>by </b><i><a class="txt" href="gfacirc1.html#E2:35"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="gfacirc1.html#E5:35"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="gfacirc1.html#E6:35"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="gfacirc1.html#T23" target="_self" title="GFACIRC1:th.23">Th23</a></i>;<br/>


</div>
