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

<b>set </b><font color="Maroon">f1</font> =  <a href="gfacirc1.html#K3" target="_self">and2c</a> ;<br/>
<b>set </b><font color="Maroon">f2</font> =  <a href="twoscomp.html#K3">and2a</a> ;<br/>
<b>set </b><font color="Maroon">f3</font> =  <a href="twoscomp.html#K2">and2</a> ;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:77"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <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="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <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#K2">and2</a> </span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <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="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span> )
 ;<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="gfacirc1.html#K3" target="_self">and2c</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="twoscomp.html#K3">and2a</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#K2">and2</a> </span><a href="tarski.html#K4">]</a></span>;<br/>
<b>let </b><font color="Maroon">s</font> be   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K20" target="_self">GFA1CarryCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>;<br/><b>let </b><font color="Maroon">a1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="circcomb.html#K10">BOOLEAN</a> , <font color="Maroon">a2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="circcomb.html#K10">BOOLEAN</a> , <font color="Maroon">a3</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="circcomb.html#K10">BOOLEAN</a> ;<br/>



<b>assume </b><a NAME="E2:77"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 ;<br/>

<b>set </b><font color="Maroon">S</font> =  <a href="gfacirc1.html#K19" target="_self">GFA1CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="msafree2.html#NM1">Vertex</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K19" target="_self">GFA1CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <b>by </b><i><a class="ref" href="gfacirc1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:77"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K19" target="_self">GFA1CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K19" target="_self">GFA1CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K19" target="_self">GFA1CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="gfacirc1.html#T11" target="_self">Th11</a>, <a class="ref" href="gfacirc1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:77"/><b>then </b><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="twoscomp.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="twoscomp.html#K1">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="twoscomp.html#K1">.</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="circuit2.html#D5">CIRCUIT2:def 5</a></i>;<br/>
<a NAME="E6:77"/><i><font color="Green">E35</font></i>: 
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,2 <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T15">FACIRC_1:15</a></i>;<br/>
<a NAME="E7:77"/>
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <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="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">a2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <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="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">a2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a3</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <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#K2">and2</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a1</font> )
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E17"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="gfacirc1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E8:77"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,2</span>)</span> <a href="twoscomp.html#K1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K21" target="_self">GFA1CarryOutput</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"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">a2</font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">a2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a3</font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="gfacirc1.html#T4" target="_self">Th4</a></i>;<br/>

<b>thus </b><a NAME="E9:77"/>
( <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <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="gfacirc1.html#K3" target="_self">and2c</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">a2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <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="twoscomp.html#K3">and2a</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <font color="Maroon">a2</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <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#K2">and2</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a1</font> )
 <b>by </b><i><a class="txt" href="gfacirc1.html#E17"><i><font color="Green">Lemma49</font></i></a>, , , <a class="ref" href="gfacirc1.html#D4" target="_self">Def4</a></i>;<br/>


</div>
