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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>

<b>defpred </b><font color="Maroon">S<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>means </b>( ( <span class="p1">(<span class="default">$1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a>  &amp; <span class="p1">(<span class="default">$1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>  &amp;  <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">$1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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>  ) or (  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">$1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <span class="p1">(<span class="default">$1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K4" title="FACIRC_1:func.4">or3</a>  &amp;  <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">$1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <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>  ) );<br/>
<a NAME="E1:40"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<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></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>  &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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>  &amp; 0 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<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></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="facirc_2.html#T8" target="_self" title="FACIRC_2:th.8">Th8</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E2:40"/><b>then </b><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="funct_5.html#T21" title="FUNCT_5:th.21">FUNCT_5:21</a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<div><i><font color="Green" title="E25">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">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>assume </b><a NAME="E1:40_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ;<br/><b>set </b><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c3">n</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">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">x</font>,<font color="Maroon" title="c2">y</font> = 
 <a href="facirc_1.html#K21" title="FACIRC_1:func.21">MajorityOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <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">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <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="c3">n</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span>
<b>by </b><i><a class="ref" href="facirc_2.html#T13" target="_self" title="FACIRC_2:th.13">Th13</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K7" title="FACIRC_1:func.7">&lt;*</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">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">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">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">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">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">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="c3">n</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></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="c3">n</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">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="facirc_1.html#K7" title="FACIRC_1:func.7">*&gt;</a></span>,<a href="facirc_1.html#K4" title="FACIRC_1:func.4">or3</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/><a NAME="E3:40_1"/><b>then </b>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="facirc_1.html#K4" title="FACIRC_1:func.4">or3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <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>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="facirc_1.html#K7" title="FACIRC_1:func.7">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">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">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">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="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">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="c3">n</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></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="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">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="facirc_1.html#K7" title="FACIRC_1:func.7">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">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">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K4" title="FACIRC_1:func.4">or3</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E4:40_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a>, <a class="ref" href="funct_5.html#T21" title="FUNCT_5:th.21">FUNCT_5:21</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E4:40"/>
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">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="facirc_2.html#E2:40"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="facirc_2.html#E3:40"><i><font color="Green" title="E25">A2</font></i></a>);</i><br/>


</div>
