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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">E38</font></i>: 
 <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>







<b>assume </b><a NAME="E2:62"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E4:62"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="facirc_1.html#E2:62"><i><font color="Green">E39</font></i></a>, <a class="ref" href="facirc_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:62"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="facirc_1.html#T12" target="_self">Th12</a></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="facirc_1.html#E1:62"><i><font color="Green">E38</font></i></a>, <a class="txt" href="facirc_1.html#E1:62_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="facirc_1.html#T29" target="_self">Th29</a></i>
<br/>.= 
 <a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="facirc_1.html#T12" target="_self">Th12</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:62"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="facirc_1.html#E4:62"><i><font color="Green">E40</font></i></a>, <a class="txt" href="facirc_1.html#E5:62"><i><font color="Green">E41</font></i></a>);</i><br/>


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