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<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/>

<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E10">A1</font></i>: 
(  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">S1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c2">S2</font> &amp;  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c2">S2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon" title="c1">S1</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/>

<b>assume </b><a NAME="E2:13"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S1</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <font color="Maroon" title="c2">S2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">A1</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">S1</font>;<br/><b>let </b><font color="Maroon" title="c5">A2</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c2">S2</font>;<br/>



<b>assume </b><a NAME="E3:13"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c4">A1</font> <a href="circcomb.html#R2" title="CIRCCOMB:pred.2">tolerates</a> <font color="Maroon" title="c5">A2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">A</font> be  <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c3">S</font>;<br/>

<b>assume </b><a NAME="E4:13"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A1</font> <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon" title="c5">A2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c6">A</font>;<br/><b>let </b><font color="Maroon" title="c8">s1</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c4">A1</font>;<br/>



<b>assume </b><a NAME="E5:13"/><i><font color="Green" title="E14">A5</font></i>: 
( <font color="Maroon" title="c8">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> &amp; <font color="Maroon" title="c8">s1</font> is <a href="circcmb3.html#V1" title="CIRCCMB3:attr.1">stabilizing</a> )
 ;<br/>

<b>let </b><font color="Maroon" title="c9">s2</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c5">A2</font>;<br/>

<b>assume </b><a NAME="E6:13"/><i><font color="Green" title="E15">A6</font></i>: 
( <font color="Maroon" title="c9">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font> &amp; <font color="Maroon" title="c9">s2</font> is <a href="circcmb3.html#V1" title="CIRCCMB3:attr.1">stabilizing</a> )
 ;<br/>

<a NAME="E7:13"/><i><font color="Green" title="E16">A7</font></i>: 
<font color="Maroon" title="c7">s</font> is <a href="circcmb3.html#V1" title="CIRCCMB3:attr.1">stabilizing</a>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="circcmb3.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="circcmb3.html#E3:13"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="circcmb3.html#E4:13"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="circcmb3.html#E5:13"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="circcmb3.html#E6:13"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="circcmb3.html#T8" target="_self" title="CIRCCMB3:th.8">Th8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">st1</font> =  <a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c8">s1</font>;<br/>
<b>set </b><font color="Maroon" title="c11">st2</font> =  <a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c9">s2</font>;<br/>
<a NAME="E8:13"/>
(  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c8">s1</font>,<span class="p1">(<span class="default"><a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c8">s1</font></span>)</span> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> &amp;  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c9">s2</font>,<span class="p1">(<span class="default"><a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c9">s2</font></span>)</span> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> )
 
<b>by </b><i><a class="txt" href="circcmb3.html#E5:13"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="circcmb3.html#E6:13"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="circcmb3.html#D5" target="_self" title="CIRCCMB3:def.5">Def5</a></i>;<br/>
<a NAME="E9:13"/><b>then </b><i><font color="Green" title="E17">A8</font></i>: 
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <span class="p2">(<span class="default"><a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c8">s1</font></span>)</span>,<span class="p2">(<span class="default"><a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c9">s2</font></span>)</span></span>)</span> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>
 
<b>by </b><i><a class="txt" href="circcmb3.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="circcmb3.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="circcmb3.html#E3:13"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="circcmb3.html#E4:13"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="circcmb3.html#E5:13"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="circcmb3.html#E6:13"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="circcmb2.html#T23" title="CIRCCMB2:th.23">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" title="c12">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:13_1"/><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c8">s1</font></span>)</span>,<span class="p1">(<span class="default"><a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c9">s2</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" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c8">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c9">s2</font> or  <a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c9">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c8">s1</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" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c8">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c9">s2</font>
 ;<br/><div class="add"><a NAME="E2:13_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c9">s2</font>
 <b>by </b><i><a class="txt" href="circcmb3.html#E1:13_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:13_1_1_1"/><b>then </b>
not  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c9">s2</font>,<font color="Maroon" title="c12">n</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb3.html#E6:13"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="circcmb3.html#D5" target="_self" title="CIRCCMB3:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E4:13_1_1_1"/>
not  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c12">n</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb3.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="circcmb3.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="circcmb3.html#E3:13"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="circcmb3.html#E4:13"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="circcmb3.html#E5:13"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="circcmb3.html#E6:13"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="circcmb2.html#T24" title="CIRCCMB2:th.24">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" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c9">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c8">s1</font>
 ;<br/><div class="add"><a NAME="E2:13_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c8">s1</font>
 <b>by </b><i><a class="txt" href="circcmb3.html#E1:13_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:13_1_1_2"/><b>then </b>
not  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c12">n</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb3.html#E5:13"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="circcmb3.html#D5" target="_self" title="CIRCCMB3:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E4:13_1_1_2"/>
not  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c12">n</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>
 <b>by </b><i><a class="txt" href="circcmb3.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="circcmb3.html#E2:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="circcmb3.html#E3:13"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="circcmb3.html#E4:13"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="circcmb3.html#E5:13"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="circcmb3.html#E6:13"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="circcmb2.html#T24" title="CIRCCMB2:th.24">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" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <span class="p1">(<span class="default"><a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c8">s1</font></span>)</span>,<span class="p1">(<span class="default"><a href="circcmb3.html#K2" title="CIRCCMB3:func.2">stabilization-time</a> <font color="Maroon" title="c9">s2</font></span>)</span>
 <b>by </b><i><a class="txt" href="circcmb3.html#E7:13"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="circcmb3.html#E9:13"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="circcmb3.html#D5" target="_self" title="CIRCCMB3:def.5">Def5</a></i>;<br/>


</div>
