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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="twoscomp.html#K2">and2</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="twoscomp.html#K2">and2</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="twoscomp.html#K2">and2</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="twoscomp.html#K25">or3</a> ;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></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">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="twoscomp.html#K2">and2</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>7</sub></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>set </b><font color="Maroon">c<sub>10</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>7</sub></font>,<font color="Maroon">c<sub>5</sub></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>set </b><font color="Maroon">c<sub>11</sub></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">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="twoscomp.html#K2">and2</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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>,<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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></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></span><a href="facirc_1.html#K7">*&gt;</a></span>,<a href="twoscomp.html#K25">or3</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="gfacirc1.html#K7" target="_self">GFA0CarryStr</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="gfacirc1.html#K5" target="_self">GFA0CarryIStr</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E1:31"/><i><font color="Green">E19</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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</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="E2:31"/><b>then </b>
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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</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="E3:31"/><b>then </b><i><font color="Green">E20</font></i>: 
not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K5" target="_self">GFA0CarryIStr</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> is <a href="facirc_1.html#V2">with_pair</a>
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="ref" href="facirc_1.html#T9">FACIRC_1:9</a></i>;<br/>
<a NAME="E4:31"/>
 <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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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>,<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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></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></span><a href="facirc_1.html#K7">*&gt;</a></span>,<a href="twoscomp.html#K25">or3</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="E5:31"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K7" target="_self">GFA0CarryStr</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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#K5" target="_self">GFA0CarryIStr</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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>,<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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></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></span><a href="facirc_1.html#K7">*&gt;</a></span>,<a href="twoscomp.html#K25">or3</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#K5" target="_self">GFA0CarryIStr</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="facirc_1.html#T6">FACIRC_1:6</a></i>;<br/>
<b>given </b><font color="Maroon">c<sub>17</sub></font> being  <a href="facirc_1.html#V1">pair</a>  <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E6:31"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>17</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#K7" target="_self">GFA0CarryStr</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D2">FACIRC_1:def 2</a><br/>

<a NAME="E7:31"/><i><font color="Green">E23</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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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>,<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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></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></span><a href="facirc_1.html#K7">*&gt;</a></span>,<a href="twoscomp.html#K25">or3</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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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>,<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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></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></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="E8:31"/><i><font color="Green">E24</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">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="twoscomp.html#K2">and2</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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></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></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="E9:31"/>
(  <a href="circcomb.html#K7">1GateCircStr</a> <span class="p1"><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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</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">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="twoscomp.html#K2">and2</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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</a>  )
 
<b>by </b><i><a class="ref" href="circcomb.html#T55">CIRCCOMB:55</a></i>;<br/>
<a NAME="E10:31"/><b>then </b><i><font color="Green">E25</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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="ref" href="circcomb.html#T15">CIRCCOMB:15</a></i>;<br/>
<a NAME="E11:31"/>
<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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K2">and2</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#K5" target="_self">GFA0CarryIStr</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></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></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="gfacirc1.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="gfacirc1.html#E10:31"><i><font color="Green">E25</font></i></a>, <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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></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></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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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>,<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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></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></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="E13:31"/><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">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="twoscomp.html#K2">and2</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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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>,<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">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></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></span><a href="facirc_1.html#K7">*&gt;</a></span>,<a href="twoscomp.html#K25">or3</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#K5" target="_self">GFA0CarryIStr</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E7:31"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E14:31"/>
contradiction
 <b>by </b><i><a class="txt" href="gfacirc1.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gfacirc1.html#E5:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gfacirc1.html#E6:31"><i><font color="Green">E22</font></i></a>, <a class="ref" href="facirc_1.html#D2">FACIRC_1:def 2</a></i>;<br/>


</div>
