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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="twoscomp.html#K3">and2a</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="gfacirc1.html#K3" target="_self">and2c</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="twoscomp.html#K4">and2b</a> ;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:123"/><i><font color="Green">E75</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>5</sub></font>,<font color="Maroon">c<sub>6</sub></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> &amp; <font color="Maroon">c<sub>5</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>6</sub></font>,<font color="Maroon">c<sub>4</sub></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> &amp; <font color="Maroon">c<sub>6</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>7</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>8</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>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>9</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>6</sub></font>,<font color="Maroon">c<sub>4</sub></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>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K32" target="_self">GFA2CarryCirc</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</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="E2:123"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="gfacirc1.html#K31" target="_self">GFA2CarryStr</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="msafree2.html#NM1">Vertex</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K31" target="_self">GFA2CarryStr</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <b>by </b><i><a class="ref" href="gfacirc1.html#T92" target="_self">Th92</a></i>;<br/>
<a NAME="E4:123"/>
( <font color="Maroon">c<sub>4</sub></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#K31" target="_self">GFA2CarryStr</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></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#K31" target="_self">GFA2CarryStr</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></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#K31" target="_self">GFA2CarryStr</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E1:123"><i><font color="Green">E75</font></i></a>, <a class="ref" href="gfacirc1.html#T94" target="_self">Th94</a></i>;<br/>
<a NAME="E5:123"/><b>then </b><i><font color="Green">E77</font></i>: 
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="twoscomp.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="twoscomp.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="twoscomp.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="circuit2.html#D5">CIRCUIT2:def 5</a></i>;<br/>
<a NAME="E6:123"/><i><font color="Green">E78</font></i>: 
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T15">FACIRC_1:15</a></i>;<br/>
<a NAME="E7:123"/>
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>10</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>4</sub></font>,<font color="Maroon">c<sub>5</sub></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#K11">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>10</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>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>12</sub></font> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>10</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>6</sub></font>,<font color="Maroon">c<sub>4</sub></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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E2:123"><i><font color="Green">E76</font></i></a>, <a class="ref" href="gfacirc1.html#T96" target="_self">Th96</a></i>;<br/>
<b>hence </b><a NAME="E8:123"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>10</sub></font>,2</span>)</span> <a href="twoscomp.html#K1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K33" target="_self">GFA2CarryOutput</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K11">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="margrel1.html#K11">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="margrel1.html#K12">'&amp;'</a> <span class="p4">(<span class="default"><a href="margrel1.html#K11">'not'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K11">'not'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p3">(<span class="default"><a href="margrel1.html#K11">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="gfacirc1.html#E6:123"><i><font color="Green">E78</font></i></a>, <a class="ref" href="gfacirc1.html#T97" target="_self">Th97</a></i>;<br/>

<b>thus </b><a NAME="E9:123"/>
( <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>10</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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#K11">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>10</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>12</sub></font> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="gfacirc1.html#E2:123"><i><font color="Green">E76</font></i></a>, <a class="txt" href="gfacirc1.html#E5:123"><i><font color="Green">E77</font></i></a>, <a class="txt" href="gfacirc1.html#E6:123"><i><font color="Green">E78</font></i></a>, <a class="ref" href="gfacirc1.html#T96" target="_self">Th96</a></i>;<br/>


</div>
