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

<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">c</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48"/><i><font color="Green">E25</font></i>: 
( <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">c</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="facirc_1.html#K3">'&amp;'</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">c</font>,<font color="Maroon">x</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="facirc_1.html#K3">'&amp;'</a> </span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c</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="facirc_1.html#K3">'&amp;'</a> </span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c</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="facirc_1.html#K1">'xor'</a> </span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="facirc_1.html#K1">'xor'</a> ;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="facirc_1.html#K23">BitAdderWithOverflowStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">S1</font> =  <a href="facirc_1.html#K12">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> ;<br/>
<b>set </b><font color="Maroon">S2</font> =  <a href="facirc_1.html#K19">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="facirc_1.html#K24">BitAdderWithOverflowCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">A1</font> =  <a href="facirc_1.html#K17">BitAdderCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">A2</font> =  <a href="facirc_1.html#K22">MajorityCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</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="facirc_1.html#K24">BitAdderWithOverflowCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span>;<br/><b>let </b><font color="Maroon">a1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="circcomb.html#K10">BOOLEAN</a> , <font color="Maroon">a2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="circcomb.html#K10">BOOLEAN</a> , <font color="Maroon">a3</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="circcomb.html#K10">BOOLEAN</a> ;<br/>



<b>assume </b><a NAME="E2:48"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E3:48"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</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">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </span>)</span> &amp; <font color="Maroon">y</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">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </span>)</span> &amp; <font color="Maroon">c</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">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="facirc_1.html#T60">FACIRC_1:60</a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</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#K19">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> &amp; <font color="Maroon">y</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#K19">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> &amp; <font color="Maroon">c</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#K19">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="facirc_1.html#T72">FACIRC_1:72</a></i>;<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="facirc_1.html#K12">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </span>)</span> as   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K17">BitAdderCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</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="facirc_1.html#K19">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> as   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K22">MajorityCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</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="facirc_1.html#K17">BitAdderCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <a href="circcomb.html#K4">+*</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K22">MajorityCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span></span>)</span> ;<br/>
<a NAME="E8:48"/>
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </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">c</font></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="facirc_1.html#T57">FACIRC_1:57</a></i>;<br/>
<a NAME="E9:48"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:48"/><i><font color="Green">E58</font></i>: 
(  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </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="facirc_1.html#K12">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </span>)</span> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</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="facirc_1.html#K19">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E11:48"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s1</font> <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">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/>
<a NAME="E12:48"/><b>then </b>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E13:48"/><b>then </b>
( <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">t</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K13">2GatesCircOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </span>)</span> <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="facirc_1.html#K15">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K13">2GatesCircOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,2</span>)</span> <a href="facirc_1.html#K15">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K13">2GatesCircOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1">'xor'</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">a2</font></span>)</span> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">a3</font> )
 
<b>by </b><i>, <a class="ref" href="facirc_2.html#T5" target="_self">Th5</a>, , <a class="ref" href="facirc_1.html#T32">FACIRC_1:32</a>, <a class="ref" href="facirc_1.html#T64">FACIRC_1:64</a></i>;<br/>
<b>hence </b><a NAME="E14:48"/>
<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> <span class="p1">(<span class="default"><a href="facirc_1.html#K16">BitAdderOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">a2</font></span>)</span> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">a3</font>
 ;<br/>

<a NAME="E15:48"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s2</font> <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#K19">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/>
<a NAME="E16:48"/><b>then </b>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="txt" href="facirc_2.html#E2:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E17:48"/><b>then </b>
( <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">t</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K21">MajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <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="facirc_1.html#K15">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K21">MajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,2</span>)</span> <a href="facirc_1.html#K15">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K21">MajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a2</font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a3</font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a1</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="facirc_2.html#T5" target="_self">Th5</a>, , , <a class="ref" href="facirc_1.html#T33">FACIRC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E18:48"/>
<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> <span class="p1">(<span class="default"><a href="facirc_1.html#K21">MajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a2</font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a3</font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a1</font></span>)</span>
 ;<br/>


</div>
