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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <a href="circcomb.html#K10">BOOLEAN</a> , <a href="circcomb.html#K10">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K14" target="_self">2GatesCircuit</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:99"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>1</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" target="_self">&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" target="_self">*&gt;</a></span>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:99_1"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2</span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:99_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="msafree2.html#NM1">Vertex</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> ;<br/><a NAME="E4:99_1"/><i><font color="Green">E64</font></i>: 
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="facirc_1.html#E1:99"><i><font color="Green">E63</font></i></a>, <a class="ref" href="facirc_1.html#T57" target="_self">Th57</a></i>;<br/><a NAME="E5:99_1"/><i><font color="Green">E65</font></i>: 
 <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <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" target="_self">&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" target="_self">*&gt;</a></span>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="facirc_1.html#K13" target="_self">2GatesCircOutput</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="facirc_1.html#T56" target="_self">Th56</a></i>;<br/><a NAME="E6:99_1"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E7:99_1"/><i><font color="Green">E67</font></i>: 
( <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</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" target="_self">&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" target="_self">*&gt;</a></span>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2</span>)</span> <a href="facirc_1.html#K15" target="_self">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K13" target="_self">2GatesCircOutput</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="facirc_1.html#K6" target="_self">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="facirc_1.html#K6" target="_self">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2</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>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2</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>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="facirc_1.html#E1:99"><i><font color="Green">E63</font></i></a>, <a class="ref" href="facirc_1.html#T62" target="_self">Th62</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="facirc_1.html#E6:99_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_1_1"><b>suppose </b></a><a NAME="E1:99_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:99_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="facirc_1.html#E4:99_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:99_1_1_1"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="facirc_1.html#E1:99"><i><font color="Green">E63</font></i></a>, <a class="txt" href="facirc_1.html#E77"><i><font color="Green">Lemma60</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_1_2"><b>suppose </b></a><a NAME="E1:99_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:99_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" target="_self">&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" target="_self">*&gt;</a></span>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K13" target="_self">2GatesCircOutput</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="facirc_1.html#E5:99_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:99_1_1_2"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="facirc_1.html#E1:99"><i><font color="Green">E63</font></i></a>, <a class="txt" href="facirc_1.html#E7:99_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="facirc_1.html#E77"><i><font color="Green">Lemma60</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:99"/>
 <a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2 <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,2</span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circuit2.html#D6">CIRCUIT2:def 6</a><br/>


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