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<b>let </b><font color="Maroon" title="c1">IIG</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</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="msafree2.html#V5" title="MSAFREE2:attr.5">monotonic</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon" title="c2">SCS</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">IIG</font>;<br/><b>let </b><font color="Maroon" title="c3">InpFs</font> be   <a href="circuit1.html#NM2" title="CIRCUIT1:NM.2">InputFuncs</a> of <font color="Maroon" title="c2">SCS</font>;<br/><b>let </b><font color="Maroon" title="c4">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font>;<br/><b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>









<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E22">A1</font></i>: 
(  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c3">InpFs</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 implies <span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 );<br/>
<a NAME="E2:25"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/>
<div><i><font color="Green" title="E24">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font>]
 ;<br/><b>thus </b><a NAME="E2:25_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<a NAME="E2:25_1_1"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c1">IIG</font> is <a href="msafree2.html#V1" title="MSAFREE2:attr.1">with_input_V</a>
 
<b>by </b><i><a class="txt" href="circuit2.html#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="msafree2.html#D4" title="MSAFREE2:def.4">MSAFREE2:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">iv</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 as    <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</font> <b>by </b><i><a class="ref" href="circuit1.html#T2" title="CIRCUIT1:th.2">CIRCUIT1:2</a></i>;<br/>
<a NAME="E4:25_1_1"/><i><font color="Green" title="E28">A7</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="circuit2.html#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="pre_circ.html#T8" title="PRE_CIRC:th.8">PRE_CIRC:8</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="circuit1.html#K2" title="CIRCUIT1:func.2">-th_InputValues</a> <font color="Maroon" title="c3">InpFs</font> = 
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="circuit2.html#E2:25_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="circuit1.html#D2" title="CIRCUIT1:def.2">CIRCUIT1:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c7">iv</font>
<b>by </b><i><a class="txt" href="circuit2.html#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="circuit2.html#E4:25_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>
;<br/>
<a NAME="E6:25_1_1"/><b>then </b><i><font color="Green" title="E29">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="circuit1.html#K2" title="CIRCUIT1:func.2">-th_InputValues</a> <font color="Maroon" title="c3">InpFs</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="circuit2.html#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="circuit2.html#T14" target="_self" title="CIRCUIT2:th.14">Th14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">Ckm</font> = <span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">m</font> as   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_2"/>
( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> or <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="circuit2.html#E1:25_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2_1"><b>suppose </b></a><a NAME="E1:25_1_1_2_1"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> = 
 <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c8">Ckm</font>
<b>by </b><i><a class="txt" href="circuit2.html#E1:25"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="circuit2.html#E1:25_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="circuit2.html#D6" target="_self" title="CIRCUIT2:def.6">Def6</a></i>
<br/>.= 
 <a href="circuit2.html#K7" title="CIRCUIT2:func.7">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">m</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="circuit1.html#K2" title="CIRCUIT1:func.2">-th_InputValues</a> <font color="Maroon" title="c3">InpFs</font></span>)</span>
<b>by </b><i><a class="txt" href="circuit2.html#E6:25_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="funct_4.html#T79" title="FUNCT_4:th.79">FUNCT_4:79</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="circuit2.html#D9" target="_self" title="CIRCUIT2:def.9">Def9</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2_2"><b>suppose </b></a><a NAME="E1:25_1_1_2_2"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E4:25"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="circuit2.html#E2:25"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="circuit2.html#E3:25"><i><font color="Green" title="E24">A3</font></i></a>);</i><br/>


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