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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="gfacirc1.html#K4" target="_self">xor2c</a> ;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:88"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K4" target="_self">xor2c</a> </span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K4" target="_self">xor2c</a> </span><a href="tarski.html#K4">]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="gfacirc1.html#K23" target="_self">GFA1AdderCirc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K23" target="_self">GFA1AdderCirc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="circcomb.html#K10">BOOLEAN</a> ;<br/>



<b>assume </b><a NAME="E2:88"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E3:88"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K4" target="_self">xor2c</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <a href="gfacirc1.html#K4" target="_self">xor2c</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E1:88"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gfacirc1.html#E2:88"><i><font color="Green">E56</font></i></a>, <a class="txt" href="gfacirc1.html#E19"><i><font color="Green">Lemma12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:88"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="gfacirc1.html#K4" target="_self">xor2c</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K4">'xor'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="gfacirc1.html#D4" target="_self">Def4</a></i>;<br/>

<b>thus </b><a NAME="E5:88"/>
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="gfacirc1.html#E1:88"><i><font color="Green">E55</font></i></a>, <a class="txt" href="gfacirc1.html#E2:88"><i><font color="Green">E56</font></i></a>, <a class="txt" href="gfacirc1.html#E19"><i><font color="Green">Lemma12</font></i></a></i>;<br/>


</div>
