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

<b>set </b><font color="Maroon">f0</font> =  <a href="twoscomp.html#K14">xor2</a> ;<br/>
<b>set </b><font color="Maroon">f1</font> =  <a href="twoscomp.html#K2">and2</a> ;<br/>
<b>set </b><font color="Maroon">f2</font> =  <a href="twoscomp.html#K2">and2</a> ;<br/>
<b>set </b><font color="Maroon">f3</font> =  <a href="twoscomp.html#K2">and2</a> ;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:59"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K14">xor2</a> </span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</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> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</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> &amp; <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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">S</font> =  <a href="gfacirc1.html#K13" target="_self">BitGFA0Str</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">S1</font> =  <a href="gfacirc1.html#K10" target="_self">GFA0AdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">S2</font> =  <a href="gfacirc1.html#K7" target="_self">GFA0CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="gfacirc1.html#K14" target="_self">BitGFA0Circ</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">A1</font> =  <a href="gfacirc1.html#K11" target="_self">GFA0AdderCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">A2</font> =  <a href="gfacirc1.html#K8" target="_self">GFA0CarryCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>;<br/>
<b>let </b><font color="Maroon">s</font> be   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K14" target="_self">BitGFA0Circ</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K10" target="_self">GFA0AdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> as   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K11" target="_self">GFA0AdderCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <b>by </b><i><a class="ref" href="facirc_1.html#T26">FACIRC_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s2</font> = <font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K7" target="_self">GFA0CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> as   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K8" target="_self">GFA0CarryCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <b>by </b><i><a class="ref" href="facirc_1.html#T26">FACIRC_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">s</font> as   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gfacirc1.html#K11" target="_self">GFA0AdderCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="circcomb.html#K4">+*</a> <span class="p2">(<span class="default"><a href="gfacirc1.html#K8" target="_self">GFA0CarryCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> ;<br/>
<a NAME="E5:59"/>
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K10" target="_self">GFA0AdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E10:59"><i><font color="Green">E35</font></i></a>, <a class="ref" href="facirc_1.html#T57">FACIRC_1:57</a></i>;<br/>
<a NAME="E6:59"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K10" target="_self">GFA0AdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</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">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E10:59"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E7:59"/>
(  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K10" target="_self">GFA0AdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K10" target="_self">GFA0AdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K7" target="_self">GFA0CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="xboole_0.html#R1">misses</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">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E8:59"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">t</font>,2</span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K10" target="_self">GFA0AdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> &amp;  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">t</font>,3</span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K10" target="_self">GFA0AdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> &amp;  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">t</font>,2</span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K7" target="_self">GFA0CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> &amp;  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">t</font>,3</span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K7" target="_self">GFA0CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="gfacirc1.html#D1" target="_self">Def1</a>, <a class="ref" href="facirc_1.html#T30">FACIRC_1:30</a>, <a class="ref" href="facirc_1.html#T31">FACIRC_1:31</a></i>;<br/>
<a NAME="E9:59"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,2 is <a href="circuit2.html#V1">stable</a>
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E10:59"><i><font color="Green">E35</font></i></a>, <a class="ref" href="facirc_1.html#T63">FACIRC_1:63</a></i>;<br/>
<b>then </b><i><font color="Green">E35</font></i>:  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,2 = 
 <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,2</span>)</span>
<b>by </b><i><a class="ref" href="circuit2.html#D6">CIRCUIT2:def 6</a></i>
<br/>.= 
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="facirc_1.html#T12">FACIRC_1:12</a></i>
;<br/>
<a NAME="E11:59"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,2 is <a href="circuit2.html#V1">stable</a>
 
<b>by </b><i><a class="txt" href="gfacirc1.html#E10:59"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>then </b><i><font color="Green">E54</font></i>:  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,2 = 
 <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,2</span>)</span>
<b>by </b><i><a class="ref" href="circuit2.html#D6">CIRCUIT2:def 6</a></i>
<br/>.= 
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="facirc_1.html#T12">FACIRC_1:12</a></i>
;<br/>
<a NAME="E13:59"/><i><font color="Green">E55</font></i>: 
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> <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">Following</a> <font color="Maroon">s</font>,2</span>)</span>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T12">FACIRC_1:12</a></i>;<br/>
<a NAME="E14:59"/><i><font color="Green">E56</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</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="gfacirc1.html#K13" target="_self">BitGFA0Str</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,3</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="gfacirc1.html#K13" target="_self">BitGFA0Str</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</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="gfacirc1.html#K10" target="_self">GFA0AdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</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="gfacirc1.html#K7" target="_self">GFA0CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/>
<a NAME="E15:59"/><i><font color="Green">E57</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" target="_self">BitGFA0Str</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></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="gfacirc1.html#K10" target="_self">GFA0AdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K7" target="_self">GFA0CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="circcomb.html#D2">CIRCCOMB:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:59_3"/>
<font color="Maroon">a</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="gfacirc1.html#K13" target="_self">BitGFA0Str</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 ;<br/><a NAME="E2:59_3"/><b>then </b>
( <font color="Maroon">a</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="gfacirc1.html#K10" target="_self">GFA0AdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> or <font color="Maroon">a</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="gfacirc1.html#K7" target="_self">GFA0CarryStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> )
 <b>by </b><i><a class="ref" href="gfacirc1.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:59_3"/><b>then </b>
( ( <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,3</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,3</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> ) or ( <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,3</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,3</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> ) )
 <b>by </b><i>, , , <a class="ref" href="gfacirc1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E4:59_3"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</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">Following</a> <font color="Maroon">s</font>,2</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="facirc_1.html#T12">FACIRC_1:12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:59"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</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">Following</a> <font color="Maroon">s</font>,2</span>)</span>
 <b>by </b><i>, <a class="ref" href="gfacirc1.html#T1" target="_self">Th1</a>, <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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