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<b>let </b><font color="Maroon" title="c1">IIG</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">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>assume </b><a NAME="E1:28"/><i><font color="Green" title="E24">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> )
 ;<br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green" title="E25">A2</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="ref" href="msafree2.html#D4" title="MSAFREE2:def.4">MSAFREE2:def 4</a></i>;<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#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="E3:28"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font>
 ;<br/>

<a NAME="E4:28"/><i><font color="Green" title="E27">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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#E1:28"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="pre_circ.html#T8" title="PRE_CIRC:th.8">PRE_CIRC:8</a></i>;<br/>
<i><font color="Green" title="E28">A5</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="circuit2.html#E2:28"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="circuit1.html#D2" title="CIRCUIT1:def.2">CIRCUIT1:def 2</a></i>
<br/>.= 
<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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="circuit2.html#E1:28"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="circuit2.html#E4:28"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="funct_1.html#D16" title="FUNCT_1:def.16">FUNCT_1:def 16</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Cn</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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font> as   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font> ;<br/>
<b>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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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="txt" href="circuit2.html#E2:28"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="circuit1.html#T2" title="CIRCUIT1:th.2">CIRCUIT1:2</a></i>;<br/>
<a NAME="E8:28"/><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c7">iv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Cn</font>
 
<b>by </b><i><a class="txt" href="circuit2.html#E1:28"><i><font color="Green" title="E24">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">Cn1</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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> 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="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:28_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">IIG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Cn</font>
 <b>by </b><i><a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></i>;<br/><b>thus </b><a NAME="E2:28_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">IIG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">Cn1</font>
 <b>by </b><i><a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></i>;<br/><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E3:28_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">IIG</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">x'</font> = <font color="Maroon" title="c9">x</font> as   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font> ;<br/><a NAME="E5:28_2"/><i><font color="Green" title="E30">A7</font></i>: 
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c10">x'</font>,<font color="Maroon" title="c2">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="circuit2.html#E3:28"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="circuit1.html#T18" title="CIRCUIT1:th.18">CIRCUIT1:18</a></i>;<br/><a NAME="E6:28_2"/><b>then </b>
<font color="Maroon" title="c6">Cn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c10">x'</font>,<font color="Maroon" title="c7">iv</font>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:28"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="circuit2.html#T16" target="_self" title="CIRCUIT2:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E7:28_2"/>
<font color="Maroon" title="c6">Cn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Cn1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:28"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="circuit2.html#E5:28_2"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="circuit2.html#T16" target="_self" title="CIRCUIT2:th.16">Th16</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/></div><b>end;</b></div>
<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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>
<br/>.= 
 <a href="circuit2.html#K7" title="CIRCUIT2:func.7">Following</a> <font color="Maroon" title="c6">Cn</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="ref" href="circuit2.html#D9" target="_self" title="CIRCUIT2:def.9">Def9</a></i>
<br/>.= 
 <a href="circuit2.html#K6" title="CIRCUIT2:func.6">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="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="txt" href="circuit2.html#E5:28"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="circuit2.html#E8:28"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="funct_4.html#T79" title="FUNCT_4:th.79">FUNCT_4:79</a></i>
;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="circuit2.html#D6" target="_self" title="CIRCUIT2:def.6">CIRCUIT2:def 6</a><br/>


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