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

<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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> ;<br/>

<b>assume </b><a NAME="E1:61"/><i><font color="Green" title="E37">A1</font></i>: 
 <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c2">S2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">S1</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">A1</font> be  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">S1</font>;<br/><b>let </b><font color="Maroon" title="c4">A2</font> be  <a href="circcomb.html#V4" title="CIRCCOMB:attr.4">gate`2=den</a> <a href="circcomb.html#V6" title="CIRCCOMB:attr.6">Boolean</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">S2</font>;<br/><b>let </b><font color="Maroon" title="c5">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">A1</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c4">A2</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">s1</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c3">A1</font>;<br/>







<b>assume </b><a NAME="E2:61"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>
 ;<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"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">s</font>,$1</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c6">s1</font>,$1;<br/>
<a NAME="E3:61"/>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T11" target="_self" title="FACIRC_1:th.11">Th11</a></i>;<br/>
<a NAME="E4:61"/><b>then </b><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="facirc_1.html#E2:61"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="facirc_1.html#T11" target="_self" title="FACIRC_1:th.11">Th11</a></i>;<br/>
<a NAME="E5:61"/><i><font color="Green" title="E40">A4</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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="E1:61_1"/><i><font color="Green" title="E41">A5</font></i>: 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c7">n</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c7">n</font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>
<b>by </b><i><a class="ref" href="facirc_1.html#T12" target="_self" title="FACIRC_1:th.12">Th12</a></i>
<br/>.= 
 <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c6">s1</font>,<font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="txt" href="facirc_1.html#E1:61"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="facirc_1.html#E1:61_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="facirc_1.html#T28" target="_self" title="FACIRC_1:th.28">Th28</a></i>
<br/>.= 
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c6">s1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="facirc_1.html#T12" target="_self" title="FACIRC_1:th.12">Th12</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:61"/>
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_1.html#E4:61"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="facirc_1.html#E5:61"><i><font color="Green" title="E40">A4</font></i></a>);</i><br/>


</div>
