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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="msafree2.html#V2">Circuit-like</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="msafree2.html#V2">Circuit-like</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>assume </b><a NAME="E1:41"/><i><font color="Green">E39</font></i>: 
(  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">S2</font> )
 ;<br/>

<b>let </b><font color="Maroon">A1</font> be  <a href="msualg_1.html#V5">non-empty</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="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E2:41"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">A1</font> <a href="circcomb.html#R2">tolerates</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">A2</font> )
 ;<br/>

<b>let </b><font color="Maroon">s1</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A1</font>;<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>let </b><font color="Maroon">s</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E3:41"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">s1</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">S1</font> &amp; <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> )
 <b>and </b><br/><a NAME="E4:41"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">s1</font> is <a href="circuit2.html#V1">stable</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</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">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">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,0</span>)</span> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> = 
<font color="Maroon">s2</font>
<b>by </b><i>, <a class="ref" href="facirc_1.html#T11">FACIRC_1:11</a></i>
<br/>.= 
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,0
<b>by </b><i><a class="ref" href="facirc_1.html#T11">FACIRC_1:11</a></i>
;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </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="E1:41_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E2:41_2"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="partfun1.html#R1">tolerates</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="ref" href="circcomb.html#D3">CIRCCOMB:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">Fs1</font> = <span class="p1">(<span class="default"><a href="facirc_1.html#K9">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">S1</font> as   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A1</font> <b>by </b><i>, <a class="ref" href="circcomb.html#T33">CIRCCOMB:33</a></i>;<br/><a NAME="E4:41_2"/><i><font color="Green">E49</font></i>: 
(  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K9">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><a class="ref" href="facirc_1.html#T12">FACIRC_1:12</a></i>;<br/><a NAME="E5:41_2"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fs1</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E6:41_2"/><b>then </b>
<font color="Maroon">Fs1</font> is <a href="circuit2.html#V1">stable</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:41_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E8:41"/>
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>3</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


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