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<span class="kw">let </span><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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a><br/>  for <font color="Olive" title="b2">s</font> being   <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">2GatesCircuit</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">f</font>)</span>)</span>  st <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/> <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b2">s</font>,2) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a></span>)</span>,<a href="margrel1.html#K5" title="MARGREL1:func.5">BOOLEAN</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <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">2GatesCircuit</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span>  st <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c4">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/> <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Olive" title="b1">s</font>,2) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">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#K10" title="FACIRC_1:func.10">2GatesCircuit</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c4">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>  )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">S</font> =  <a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>);<br/>
<span class="kw">assume </span><a NAME="E1:90"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c4">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2) is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="90_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span></span>)</span> <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="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <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="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> ) )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:90_1"/>
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span></span>)</span> <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="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <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="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#T3" onmouseover="rs('circuit1/T3')" onmouseout="rh()">CIRCUIT1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><a NAME="E2:90_1"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span></span>)</span> <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="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T45" onmouseover="rs('xboole_1/T45')" onmouseout="rh()">XBOOLE_1:45</a></span>;<br/><span class="kw">assume </span><a NAME="E3:90_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">v</font> = <font color="Maroon" title="c7">a</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#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> ;<br/><a NAME="E5:90_1"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
 <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c4">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">2GatesCircOutput</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T56" target="_self" onmouseover="rs('facirc_1/T56')" onmouseout="rh()">Th56</a></span>;<br/><a NAME="E6:90_1"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="facirc_1.html#T62" target="_self" onmouseover="rs('facirc_1/T62')" onmouseout="rh()">Th62</a></span>;<br/><a NAME="E7:90_1"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
( <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="facirc_1.html#T62" target="_self" onmouseover="rs('facirc_1/T62')" onmouseout="rh()">Th62</a></span>;<br/><a NAME="E8:90_1"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
( <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</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="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c4">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="facirc_1.html#K11" title="FACIRC_1:func.11">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">2GatesCircOutput</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">c</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="facirc_1.html#T62" target="_self" onmouseover="rs('facirc_1/T62')" onmouseout="rh()">Th62</a></span>;<br/><a NAME="E9:90_1"/><span class="lab"><font color="Green" title="E61">A7</font></span>: 
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="facirc_1.html#T57" target="_self" onmouseover="rs('facirc_1/T57')" onmouseout="rh()">Th57</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:90_1_1"/>
( <font color="Maroon" title="c8">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> or <font color="Maroon" title="c8">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90_1"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:90_1_1_1"/>
<font color="Maroon" title="c8">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:90_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> or <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> or <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:90_1"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:90_1_1_1"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="facirc_1.html#E76"><span class="lab"><font color="Green" title="E52">Lm1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:90_1_1_2"/>
<font color="Maroon" title="c8">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree2.html#K3" title="MSAFREE2:func.3">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">2GatesCircStr</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:90_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c4">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">2GatesCircOutput</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_1"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:90_1_1_2"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E8:90_1"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E7:90_1"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E6:90_1"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="facirc_1.html#E76"><span class="lab"><font color="Green" title="E52">Lm1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:90"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> (<font color="Maroon" title="c5">s</font>,2)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="circuit2.html#D6" onmouseover="rs('circuit2/D6')" onmouseout="rh()">CIRCUIT2:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
