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<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>assume </b><a NAME="E1:62"/><i><font color="Green">E33</font></i>: 
 <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font>
 ;<br/>

<b>let </b><font color="Maroon">A1</font> be  <a href="circcomb.html#V4">gate`2=den</a> <a href="circcomb.html#V6">Boolean</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">A2</font> be  <a href="circcomb.html#V4">gate`2=den</a> <a href="circcomb.html#V6">Boolean</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">A2</font></span>)</span>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A2</font>;<br/>







<b>assume </b><a NAME="E2:62"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:62"/>
 <a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s</font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:62"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:62"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">n</font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>
<b>by </b><i/>
<br/>.= 
 <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="facirc_1.html#E2:62"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
 <a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s2</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:62"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="facirc_1.html#T1" target="_self">Th1</a>);</i><br/>


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