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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> ;<br/>

<b>assume </b><a NAME="E1:15"/><i><font color="Green">E34</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>
 ;<br/>

<b>let </b><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="E2:15"/><i><font color="Green">E36</font></i>: 
<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>assume </b><a NAME="E3:15"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A1</font> <a href="circcomb.html#R2">tolerates</a> <font color="Maroon">A2</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="E4:15"/><i><font color="Green">E38</font></i>: 
<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">s</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">A</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>assume </b><a NAME="E5:15"/><i><font color="Green">E45</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">s1</font> is <a href="circcmb3.html#V1" target="_self">stabilizing</a> )
 ;<br/>

<b>set </b><font color="Maroon">st1</font> =  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</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>assume </b><a NAME="E6:15"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</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> &amp; <font color="Maroon">s2</font> is <a href="circcmb3.html#V1" target="_self">stabilizing</a> )
 ;<br/>

<a NAME="E7:15"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">s</font> is <a href="circcmb3.html#V1" target="_self">stabilizing</a>
 
<b>by </b><i>, , , , , , </i>;<br/>
<b>set </b><font color="Maroon">st2</font> =  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s2</font>;<br/>
<a NAME="E8:15"/><i><font color="Green">E60</font></i>: 
(  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</font></span>)</span> is <a href="circuit2.html#V1">stable</a> &amp;  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s2</font></span>)</span> is <a href="circuit2.html#V1">stable</a> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:15"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s2</font></span>)</span></span>)</span> is <a href="circuit2.html#V1">stable</a>
 
<b>by </b><i>, , , , , , <a class="ref" href="circcmb2.html#T20">CIRCCMB2:20</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s2</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1"/>
(  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1"/>
 <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:15_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:15_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s2</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:15_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
not  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">m</font> is <a href="circuit2.html#V1">stable</a>
 <b>by </b><i>, </i>;<br/><a NAME="E7:15_1_1_1"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</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">S1</font>
 <b>by </b><i>, , , , , <a class="ref" href="circcmb2.html#T14">CIRCCMB2:14</a></i>;<br/><b>then </b> <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s2</font>,<font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<span class="p3">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</font></span>)</span></span>)</span>,<font color="Maroon">m</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>
<b>by </b><i>, , , , , , <a class="ref" href="circcmb2.html#T19">CIRCCMB2:19</a></i>
<br/>.= 
<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">S2</font>
<b>by </b><i>, <a class="ref" href="facirc_1.html#T13">FACIRC_1:13</a></i>
;<br/><b>hence </b><a NAME="E9:15_1_1_1"/>
not  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font> is <a href="circuit2.html#V1">stable</a>
 <b>by </b><i>, , , , <a class="ref" href="circcmb2.html#T18">CIRCCMB2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
not  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<font color="Maroon">n</font> is <a href="circuit2.html#V1">stable</a>
 <b>by </b><i>, </i>;<br/><a NAME="E3:15_1_1_2"/>
<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> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s1</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , <a class="ref" href="circcmb2.html#T14">CIRCCMB2:14</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_2"/>
not  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">s</font>,<font color="Maroon">n</font> is <a href="circuit2.html#V1">stable</a>
 <b>by </b><i>, , , , <a class="ref" href="circcmb2.html#T18">CIRCCMB2:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:15"/>
 <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">s2</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


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