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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="msafree2.html#V2">Circuit-like</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green">E10</font></i>: 
(  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></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="E2:13"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="msualg_1.html#V5">non-empty</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>5</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E3:13"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="circcomb.html#R2">tolerates</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E4:13"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="circcomb.html#K4">+*</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</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="E5:13"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> is <a href="circcmb3.html#V1" target="_self">stabilizing</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>assume </b><a NAME="E6:13"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</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> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="circcmb3.html#V1" target="_self">stabilizing</a> )
 ;<br/>

<a NAME="E7:13"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="circcmb3.html#V1" target="_self">stabilizing</a>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E1:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="circcmb3.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="circcmb3.html#E3:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="circcmb3.html#E4:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="circcmb3.html#E5:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="circcmb3.html#E6:13"><i><font color="Green">E15</font></i></a>, <a class="ref" href="circcmb3.html#T8" target="_self">Th8</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<a NAME="E8:13"/>
(  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> is <a href="circuit2.html#V1">stable</a> &amp;  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> is <a href="circuit2.html#V1">stable</a> )
 
<b>by </b><i><a class="txt" href="circcmb3.html#E5:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="circcmb3.html#E6:13"><i><font color="Green">E15</font></i></a>, <a class="ref" href="circcmb3.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E9:13"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> is <a href="circuit2.html#V1">stable</a>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E1:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="circcmb3.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="circcmb3.html#E3:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="circcmb3.html#E4:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="circcmb3.html#E5:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="circcmb3.html#E6:13"><i><font color="Green">E15</font></i></a>, <a class="ref" href="circcmb2.html#T23">CIRCCMB2:23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_1"/>
(  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>9</sub></font> or  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>8</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_1_1"/>
 <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><a NAME="E2:13_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="circcmb3.html#E1:13_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:13_1_1_1"/><b>then </b>
not  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font> is <a href="circuit2.html#V1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb3.html#E6:13"><i><font color="Green">E15</font></i></a>, <a class="ref" href="circcmb3.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E4:13_1_1_1"/>
not  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font> is <a href="circuit2.html#V1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb3.html#E1:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="circcmb3.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="circcmb3.html#E3:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="circcmb3.html#E4:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="circcmb3.html#E5:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="circcmb3.html#E6:13"><i><font color="Green">E15</font></i></a>, <a class="ref" href="circcmb2.html#T24">CIRCCMB2:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_1_2"/>
 <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:13_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="circcmb3.html#E1:13_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:13_1_1_2"/><b>then </b>
not  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font> is <a href="circuit2.html#V1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb3.html#E5:13"><i><font color="Green">E14</font></i></a>, <a class="ref" href="circcmb3.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E4:13_1_1_2"/>
not  <a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font> is <a href="circuit2.html#V1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb3.html#E1:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="circcmb3.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="txt" href="circcmb3.html#E3:13"><i><font color="Green">E12</font></i></a>, <a class="txt" href="circcmb3.html#E4:13"><i><font color="Green">E13</font></i></a>, <a class="txt" href="circcmb3.html#E5:13"><i><font color="Green">E14</font></i></a>, <a class="txt" href="circcmb3.html#E6:13"><i><font color="Green">E15</font></i></a>, <a class="ref" href="circcmb2.html#T24">CIRCCMB2:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:13"/>
 <a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="circcmb3.html#K2" target="_self">stabilization-time</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circcmb3.html#E7:13"><i><font color="Green">E16</font></i></a>, <a class="txt" href="circcmb3.html#E9:13"><i><font color="Green">E17</font></i></a>, <a class="ref" href="circcmb3.html#D5" target="_self">Def5</a></i>;<br/>


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