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

<b>let </b><font color="Maroon">x</font> be  non <a href="facirc_1.html#V1" target="_self">pair</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be  non <a href="facirc_1.html#V1" target="_self">pair</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> be  non <a href="facirc_1.html#V1" target="_self">pair</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="facirc_1.html#K1" target="_self">'xor'</a> ;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="facirc_1.html#K23" target="_self">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" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> ;<br/>
<b>set </b><font color="Maroon">S2</font> =  <a href="facirc_1.html#K19" target="_self">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" target="_self">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" target="_self">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" target="_self">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" target="_self">BitAdderWithOverflowCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</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="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'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" target="_self">BitAdderCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <b>by </b><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" target="_self">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" target="_self">MajorityCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <b>by </b><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" target="_self">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" target="_self">MajorityCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span></span>)</span> ;<br/>
<a NAME="E4:137"/>
( not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> is <a href="facirc_1.html#V2" target="_self">with_pair</a> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> is   <a href="relat_1.html#NM1">Relation</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> is <a href="facirc_1.html#V2" target="_self">with_pair</a> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> is   <a href="relat_1.html#NM1">Relation</a> )
 
<b>by </b><i><a class="txt" href="facirc_1.html#E13:137"><i><font color="Green">E57</font></i></a>, <a class="ref" href="facirc_1.html#T5" target="_self">Th5</a>, <a class="ref" href="facirc_1.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E5:137"/><b>then </b>
(  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'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#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" target="_self">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#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:137"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <a href="facirc_1.html#K9" target="_self">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" target="_self">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="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> &amp;  <a href="facirc_1.html#K9" target="_self">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" target="_self">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="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> &amp;  <a href="facirc_1.html#K9" target="_self">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" target="_self">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="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> &amp;  <a href="facirc_1.html#K9" target="_self">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" target="_self">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="facirc_1.html#K19" target="_self">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#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:137"/>
 <a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s1</font>,2 is <a href="circuit2.html#V1">stable</a>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T7" target="_self">Th7</a></i>;<br/>
<b>then </b><i><font color="Green">E35</font></i>:  <a href="facirc_1.html#K9" target="_self">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" target="_self">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" target="_self">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/>
;<br/>
<a NAME="E9:137"/>
 <a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s2</font>,2 is <a href="circuit2.html#V1">stable</a>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E54</font></i>:  <a href="facirc_1.html#K9" target="_self">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" target="_self">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" target="_self">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/>
;<br/>
<a NAME="E11:137"/><i><font color="Green">E55</font></i>: 
 <a href="facirc_1.html#K9" target="_self">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" target="_self">Following</a> <font color="Maroon">s</font>,2</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:137"/><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" target="_self">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="facirc_1.html#K23" target="_self">BitAdderWithOverflowStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">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="facirc_1.html#K23" target="_self">BitAdderWithOverflowStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">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="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">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="facirc_1.html#K19" target="_self">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="E13:137"/><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="facirc_1.html#K23" target="_self">BitAdderWithOverflowStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</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="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </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="facirc_1.html#K19" target="_self">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="circcomb.html#D2">CIRCCOMB:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_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="E1:137_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="facirc_1.html#K23" target="_self">BitAdderWithOverflowStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span>
 ;<br/><a NAME="E2:137_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="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </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="facirc_1.html#K19" target="_self">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_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:137_3"/><b>then </b>
( ( <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">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" target="_self">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" target="_self">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" target="_self">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" target="_self">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" target="_self">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" target="_self">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" target="_self">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="facirc_1.html#D4" target="_self">Def4</a>, <a class="ref" href="facirc_1.html#D5" target="_self">Def5</a>, <a class="ref" href="facirc_1.html#D6" target="_self">Def6</a>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E4:137_3"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">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" target="_self">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#D5" target="_self">Def5</a>, <a class="ref" href="facirc_1.html#D6" target="_self">Def6</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:137"/>
 <a href="facirc_1.html#K9" target="_self">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" target="_self">Following</a> <font color="Maroon">s</font>,2</span>)</span>
 <b>by </b><i><a class="ref" href="facirc_1.html#D7" target="_self">Def7</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/>


</div>
