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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="E1:29"/><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="E2:29"/><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">s1</font>, <font color="Maroon" title="c5">s2</font> be   <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="c6">dSCS</font> =  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Cs1</font> = <span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s1</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">dSCS</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="c8">Cs2</font> = <span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c5">s2</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">dSCS</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="c9">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#E2:29"><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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:29_1"/>
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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">Cs1</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="E2:29_1"/>
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#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">Cs2</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="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E3:29_1"/>
<font color="Maroon" title="c10">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="c11">x'</font> = <font color="Maroon" title="c10">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="E5:29_1"/>
( <font color="Maroon" title="c7">Cs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="c11">x'</font>,<font color="Maroon" title="c9">iv</font> &amp; <font color="Maroon" title="c8">Cs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="c11">x'</font>,<font color="Maroon" title="c9">iv</font> )
 <b>by </b><i><a class="txt" href="circuit2.html#E1:29"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="circuit2.html#T17" target="_self" title="CIRCUIT2:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E6:29_1"/>
<font color="Maroon" title="c7">Cs1</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> <font color="Maroon" title="c8">Cs2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:29"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c4">s1</font>,<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"><a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font></span>)</span> <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="c5">s2</font>,<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"><a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font></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/>


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