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<div class="add">

<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>for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="facirc_1.html#V3" title="FACIRC_1:attr.3">nonpair-yielding</a>  <a href="circcomb.html#M1" title="CIRCCOMB:mode.1">FinSeqLen</a> of $1 holds   <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default">$1 <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">y</font></span>)</span>;<br/>
<a NAME="E1:43"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be  <a href="facirc_1.html#V3" title="FACIRC_1:attr.3">nonpair-yielding</a>  <a href="circcomb.html#M1" title="CIRCCOMB:mode.1">FinSeqLen</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>

<b>set </b><font color="Maroon" title="c3">f</font> = <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K9" title="CIRCCOMB:func.9">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ;<br/>
<a NAME="E1:43_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="circcomb.html#D12" title="CIRCCOMB:def.12">CIRCCOMB:def 12</a></i>;<br/>
<a NAME="E2:43_1"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E3:43_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</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>  <a href="circcomb.html#K6" title="CIRCCOMB:func.6">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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K9" title="CIRCCOMB:func.9">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&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/>
<b>hence </b><a NAME="E4:43_1"/>
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="facirc_2.html#E2:43_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="circcomb.html#T49" title="CIRCCOMB:th.49">CIRCCOMB:49</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:43"/><i><font color="Green" title="E27">A3</font></i>: 
for <font color="Olive" title="b1">i</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">i</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="E1:43_2"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">i</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be  <a href="facirc_1.html#V3" title="FACIRC_1:attr.3">nonpair-yielding</a>  <a href="circcomb.html#M1" title="CIRCCOMB:mode.1">FinSeqLen</a> of <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>

<b>consider </b><font color="Maroon" title="c4">x'</font> being  <a href="facirc_1.html#V3" title="FACIRC_1:attr.3">nonpair-yielding</a>  <a href="circcomb.html#M1" title="CIRCCOMB:mode.1">FinSeqLen</a> of <font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c5">z1</font> being  non <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:43_2"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x'</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c5">z1</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="facirc_2.html#E24"><i><font color="Green" title="E15">Lm2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">y'</font> being  <a href="facirc_1.html#V3" title="FACIRC_1:attr.3">nonpair-yielding</a>  <a href="circcomb.html#M1" title="CIRCCOMB:mode.1">FinSeqLen</a> of <font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c7">z2</font> being  non <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:43_2"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y'</font> <a href="facirc_1.html#K8" title="FACIRC_1:func.8">^</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">z2</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="facirc_2.html#E24"><i><font color="Green" title="E15">Lm2</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">S</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>;<br/>
<a NAME="E6:43_2"/><i><font color="Green" title="E31">A7</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:43_2"/><i><font color="Green" title="E32">A8</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c5">z1</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp; <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c5">z1</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E8:43_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D12" title="CIRCCOMB:def.12">CIRCCOMB:def 12</a></i>;<br/>
<a NAME="E9:43_2"/><b>then </b><i><font color="Green" title="E33">A9</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z1</font>
 
<b>by </b><i><a class="txt" href="facirc_2.html#E3:43_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="facirc_2.html#E6:43_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="facirc_2.html#E7:43_2"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E10:43_2"/><i><font color="Green" title="E34">A10</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">z2</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp; <span class="p1"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">z2</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E11:43_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D12" title="CIRCCOMB:def.12">CIRCCOMB:def 12</a></i>;<br/>
<a NAME="E12:43_2"/><b>then </b><i><font color="Green" title="E35">A11</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z2</font>
 
<b>by </b><i><a class="txt" href="facirc_2.html#E5:43_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="facirc_2.html#E6:43_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="facirc_2.html#E10:43_2"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E13:43_2"/><i><font color="Green" title="E36">A12</font></i>: 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c7">z2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <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">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>,<font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c7">z2</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#T100" title="ENUMSET1:th.100">ENUMSET1:100</a></i>;<br/>
<i><font color="Green" title="E37">A13</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">x</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">x'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c5">z1</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="facirc_2.html#E3:43_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">x'</font></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"><font color="Maroon" title="c5">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>
;<br/>
<i><font color="Green" title="E38">A14</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">y</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">y'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">z2</font></span><a href="facirc_1.html#K5" title="FACIRC_1:func.5">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="facirc_2.html#E5:43_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">y'</font></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"><font color="Maroon" title="c7">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>
;<br/>
<a NAME="E16:43_2"/><i><font color="Green" title="E39">A15</font></i>: 
( <font color="Maroon" title="c5">z1</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"><font color="Maroon" title="c7">z2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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> &amp; <font color="Maroon" title="c7">z2</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">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>,<font color="Maroon" title="c5">z1</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> &amp; <font color="Maroon" title="c1">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</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"><font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c7">z2</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> &amp; <font color="Maroon" title="c1">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</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"><font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c7">z2</font></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="ref" href="facirc_2.html#T26" target="_self" title="FACIRC_2:th.26">Th26</a></i>;<br/>
<a NAME="E17:43_2"/>
( <font color="Maroon" title="c4">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x'</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y'</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E18:43_2"/><b>then </b>
<font color="Maroon" title="c1">i</font> <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c4">x'</font>,<font color="Maroon" title="c6">y'</font>
 
<b>by </b><i><a class="txt" href="facirc_2.html#E3:43_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="facirc_2.html#E5:43_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="facirc_2.html#T11" target="_self" title="FACIRC_2:th.11">Th11</a></i>;<br/>
<b>hence </b> <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="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> = 
<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="c1">i</font> <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c4">x'</font>,<font color="Maroon" title="c6">y'</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> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c7">z2</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</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="c1">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="facirc_2.html#E9:43_2"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="facirc_2.html#E12:43_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="facirc_2.html#T27" target="_self" title="FACIRC_2:th.27">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">x'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">y'</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> <font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c7">z2</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</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="c1">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="facirc_2.html#E1:43_2"><i><font color="Green" title="E28">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">x'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">y'</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"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c7">z2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="c1">i</font> <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="facirc_2.html#E16:43_2"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="facirc_2.html#T23" target="_self" title="FACIRC_2:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">x'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">y'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">z1</font>,<font color="Maroon" title="c7">z2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="txt" href="facirc_2.html#E13:43_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="enumset1.html#T136" title="ENUMSET1:th.136">ENUMSET1:136</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">x'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">y'</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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">x'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">y'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z1</font></span><a href="tarski.html#K1" title="TARSKI: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"><font color="Maroon" title="c7">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">x'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">y'</font></span>)</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"><font color="Maroon" title="c7">z2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">y</font></span>)</span>
<b>by </b><i><a class="txt" href="facirc_2.html#E14:43_2"><i><font color="Green" title="E37">A13</font></i></a>, <a class="txt" href="facirc_2.html#E15:43_2"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:43"/>
for <font color="Olive" title="b1">i</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">i</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#E1:43"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="facirc_2.html#E2:43"><i><font color="Green" title="E27">A3</font></i></a>);</i><br/>


</div>
