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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>set </b><font color="Maroon" title="c4">xy</font> = <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">S1</font> =  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">A1</font> =  <a href="facirc_1.html#K10" title="FACIRC_1:func.10">1GateCircuit</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> ;<br/>
<b>set </b><font color="Maroon" title="c7">S2</font> =  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> ;<br/>
<b>set </b><font color="Maroon" title="c8">A2</font> =  <a href="facirc_1.html#K10" title="FACIRC_1:func.10">1GateCircuit</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> ;<br/>
<b>set </b><font color="Maroon" title="c9">S3</font> =  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircStr</a> <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">x</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> ;<br/>
<b>set </b><font color="Maroon" title="c10">A3</font> =  <a href="facirc_1.html#K10" title="FACIRC_1:func.10">1GateCircuit</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">x</font>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> ;<br/>
<b>set </b><font color="Maroon" title="c11">S</font> =  <a href="facirc_1.html#K18" title="FACIRC_1:func.18">MajorityIStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<b>set </b><font color="Maroon" title="c12">A</font> =  <a href="facirc_1.html#K20" title="FACIRC_1:func.20">MajorityICirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<b>let </b><font color="Maroon" title="c13">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K20" title="FACIRC_1:func.20">MajorityICirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="circcomb.html#K10" title="CIRCCOMB:func.10">BOOLEAN</a> ;<br/>



<b>assume </b><a NAME="E1:110"/><i><font color="Green" title="E65">A1</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">y</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c16">v1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span>)</span> <b>by </b><i><a class="ref" href="facirc_1.html#T47" target="_self" title="FACIRC_1:th.47">Th47</a></i>;<br/>
<a NAME="E3:110"/><i><font color="Green" title="E66">A2</font></i>: 
 <a href="facirc_1.html#K18" title="FACIRC_1:func.18">MajorityIStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span>)</span> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span>)</span> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <span class="p2">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircStr</a> <span class="p3"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">x</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="circcomb.html#T10" title="CIRCCOMB:th.10">CIRCCOMB:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">v</font> = <font color="Maroon" title="c16">v1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K18" title="FACIRC_1:func.18">MajorityIStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> <b>by </b><i><a class="ref" href="facirc_1.html#T21" target="_self" title="FACIRC_1:th.21">Th21</a></i>;<br/>
<a NAME="E5:110"/><i><font color="Green" title="E67">A3</font></i>: 
 <a href="facirc_1.html#K20" title="FACIRC_1:func.20">MajorityICirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K10" title="FACIRC_1:func.10">1GateCircuit</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span>)</span> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K10" title="FACIRC_1:func.10">1GateCircuit</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span>)</span> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K10" title="FACIRC_1:func.10">1GateCircuit</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">x</font>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T25" target="_self" title="FACIRC_1:th.25">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">s1</font> = <font color="Maroon" title="c13">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span>)</span> as   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K10" title="FACIRC_1:func.10">1GateCircuit</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span>)</span> <b>by </b><i><a class="ref" href="facirc_1.html#T26" target="_self" title="FACIRC_1:th.26">Th26</a></i>;<br/>
<a NAME="E7:110"/><i><font color="Green" title="E68">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">xx</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c20">yy</font> = <font color="Maroon" title="c2">y</font> as   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircStr</a> <span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span>)</span> <b>by </b><i><a class="ref" href="facirc_1.html#T43" target="_self" title="FACIRC_1:th.43">Th43</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">xx</font> = <font color="Maroon" title="c19">xx</font>, <font color="Maroon" title="c22">yy</font> = <font color="Maroon" title="c20">yy</font> as   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K18" title="FACIRC_1:func.18">MajorityIStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> <b>by </b><i><a class="txt" href="facirc_1.html#E3:110"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="facirc_1.html#T20" target="_self" title="FACIRC_1:th.20">Th20</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c13">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c18">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">v</font>
<b>by </b><i><a class="txt" href="facirc_1.html#E3:110"><i><font color="Green" title="E66">A2</font></i></a>, <a class="txt" href="facirc_1.html#E5:110"><i><font color="Green" title="E67">A3</font></i></a>, <a class="ref" href="circcomb.html#T72" title="CIRCCOMB:th.72">CIRCCOMB:72</a></i>
<br/>.= 
<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">xx</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">yy</font></span>)</span></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>
<b>by </b><i><a class="ref" href="facirc_1.html#T48" target="_self" title="FACIRC_1:th.48">Th48</a></i>
<br/>.= 
<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="facirc_1.html#K15" title="FACIRC_1:func.15">.</a> <font color="Maroon" title="c21">xx</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">yy</font></span>)</span></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>
<b>by </b><i><a class="txt" href="facirc_1.html#E7:110"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="facirc_1.html#K15" title="FACIRC_1:func.15">.</a> <font color="Maroon" title="c21">xx</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">s</font> <a href="facirc_1.html#K15" title="FACIRC_1:func.15">.</a> <font color="Maroon" title="c22">yy</font></span>)</span></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>
<b>by </b><i><a class="txt" href="facirc_1.html#E7:110"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c14">a</font> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <font color="Maroon" title="c15">b</font>
<b>by </b><i><a class="txt" href="facirc_1.html#E1:110"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="facirc_1.html#D6" target="_self" title="FACIRC_1:def.6">Def6</a></i>
;<br/>


</div>
