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

<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be  <a href="facirc_1.html#V3" title="FACIRC_1:attr.3">nonpair-yielding</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b> non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="circcomb.html#V1" title="CIRCCOMB:attr.1">unsplit</a> <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</a> <a href="circcomb.html#V3" title="CIRCCOMB:attr.3">gate`2isBoolean</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  = $1 <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  =  <a href="facirc_1.html#K23" title="FACIRC_1:func.23">BitAdderWithOverflowStr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,$1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default">$1 <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span>)</span> = $1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>;<br/>
<b>consider </b><font color="Maroon" title="c3">h</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:42"/><i><font color="Green" title="E26">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" title="FACIRC_1:func.19">MajorityStr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,$1</span>)</span> =  <a href="facirc_1.html#K21" title="FACIRC_1:func.21">MajorityOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,$1;<br/>
<b>set </b><font color="Maroon" title="c4">k</font> = <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K10" title="CIRCCOMB:func.10">BOOLEAN</a> </span>)</span> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ;<br/>
<a NAME="E3:42"/><i><font color="Green" title="E27">A2</font></i>: 
0 <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircStr</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K10" title="CIRCCOMB:func.10">BOOLEAN</a> </span>)</span> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="facirc_2.html#T8" target="_self" title="FACIRC_2:th.8">Th8</a></i>;<br/>
<a NAME="E4:42"/><b>then </b><i><font color="Green" title="E28">A3</font></i>: 
 <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(0) is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T38" title="FACIRC_1:th.38">FACIRC_1:38</a></i>;<br/>
<a NAME="E5:42"/><i><font color="Green" title="E29">A4</font></i>: 
not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(0) is <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>
 
<b>by </b><i><a class="txt" href="facirc_2.html#E3:42"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="facirc_1.html#T39" title="FACIRC_1:th.39">FACIRC_1:39</a></i>;<br/>
<a NAME="E6:42"/>
<font color="Maroon">H<sub>4</sub></font>(0) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="facirc_2.html#E2:42"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E7:42"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(0)
 
;<br/>
<a NAME="E8:42"/><i><font color="Green" title="E31">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>) is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T88" title="FACIRC_1:th.88">FACIRC_1:88</a></i>;<br/>
<div><i><font color="Green" title="E32">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c5">n</font>)
 ;<br/><a NAME="E2:42_1"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c5">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="facirc_2.html#E2:42"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/><a NAME="E3:42_1"/><b>then </b>
( <font color="Maroon" title="c6">x</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="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">x</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="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="facirc_2.html#E1:42_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="facirc_2.html#T26" target="_self" title="FACIRC_2:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E4:42_1"/>
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">n</font>) <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="facirc_2.html#T23" target="_self" title="FACIRC_2:th.23">Th23</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:42"/><i><font color="Green" title="E33">A9</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/> not <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:42_2"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c5">n</font>)
 ;<br/>

<a NAME="E2:42_2"/><i><font color="Green" title="E35">A11</font></i>: 
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">n</font>) <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="facirc_2.html#E9:42"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="facirc_2.html#E1:42_2"><i><font color="Green" title="E34">A10</font></i></a></i>;<br/>
<b>thus </b><a NAME="E3:42_2"/>
not <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font> be  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D2" title="FACIRC_1:def.2">FACIRC_1:def 2</a><br/>

<b>assume </b><a NAME="E1:42_2_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:42_2_1"/><b>then </b>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">n</font>) &amp; not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:42_2_1"/><b>then </b>
( ( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> ) &amp; not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="facirc_2.html#E2:42_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:42_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<div><i><font color="Green" title="E34">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c6">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:42_3"/><i><font color="Green" title="E35">A13</font></i>: 
( <font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font>) &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> )
 ;<br/><a NAME="E2:42_3"/><b>then </b><i><font color="Green" title="E36">A14</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> &amp; <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font> )
 <b>by </b><i><a class="txt" href="facirc_2.html#E2:42"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">S</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c5">n</font>)
 <b>by </b><i><a class="txt" href="facirc_2.html#E1:42_3"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="facirc_2.html#T13" target="_self" title="FACIRC_2:th.13">Th13</a></i>;<br/><b>thus </b><a NAME="E4:42_3"/>
<font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c5">n</font>)
 <b>by </b><i><a class="txt" href="facirc_2.html#E2:42_3"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="facirc_2.html#T13" target="_self" title="FACIRC_2:th.13">Th13</a></i>;<br/><a NAME="E5:42_3"/>
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c5">n</font>) <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">x</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="facirc_2.html#E9:42"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="facirc_2.html#E1:42_3"><i><font color="Green" title="E35">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E6:42_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c5">n</font>)
 <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><a NAME="E7:42_3"/><i><font color="Green" title="E37">A15</font></i>: 
 <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c5">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><a href="facirc_1.html#K13" title="FACIRC_1:func.13">2GatesCircOutput</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">x</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">x</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c7">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K3" title="FACIRC_1:func.3">'&amp;'</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K21" title="FACIRC_1:func.21">MajorityOutput</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="facirc_2.html#T24" target="_self" title="FACIRC_2:th.24">Th24</a></i>;<br/><a NAME="E8:42_3"/>
<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c5">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c5">n</font>)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E9:42_3"/>
<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c5">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c5">n</font>)
 <b>by </b><i><a class="txt" href="facirc_2.html#E7:42_3"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:42"/><i><font color="Green" title="E35">A16</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> (  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp;  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> &amp; not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) is <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a> )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S11" title="CIRCCMB2:sch.11">CIRCCMB2:sch 11</a>(<a class="txt" href="facirc_2.html#E4:42"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="facirc_2.html#E5:42"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="facirc_2.html#E7:42"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="facirc_2.html#E8:42"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="facirc_2.html#E10:42"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="facirc_2.html#E11:42"><i><font color="Green" title="E34">A12</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E13:42"/>
<font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="facirc_2.html#E2:42"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:42"/>
(  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p3">(<span class="default"><a href="facirc_1.html#K23" title="FACIRC_1:func.23">BitAdderWithOverflowStr</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp;  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span>)</span> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> &amp; not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span>)</span> is <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a> )
 <b>by </b><i><a class="txt" href="facirc_2.html#E12:42"><i><font color="Green" title="E35">A16</font></i></a></i>;<br/>


</div>
