<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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="struct_0.html#V11" title="STRUCT_0:attr.11">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">SCS</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">IIG</font><br/>  for <font color="Olive" title="b2">InpFs</font> being   <a href="circuit1.html#NM2" title="CIRCUIT1:NM.2">InputFuncs</a> of <font color="Olive" title="b1">SCS</font>  st  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b2">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>  holds <br/> for <font color="Olive" title="b3">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b1">SCS</font><br/>  for <font color="Olive" title="b4">n</font> being    <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>   st <font color="Olive" title="b4">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="Olive" title="b1">SCS</font> holds <br/><span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> (<font color="Olive" title="b3">s</font>,<font color="Olive" title="b2">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">n</font> is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">SCS</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">IIG</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">InpFs</font> being   <a href="circuit1.html#NM2" title="CIRCUIT1:NM.2">InputFuncs</a> of <font color="Maroon" title="c2">SCS</font>  st  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b1">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>  holds <br/> for <font color="Olive" title="b2">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font><br/>  for <font color="Olive" title="b3">n</font> being    <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>   st <font color="Olive" title="b3">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> holds <br/><span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> (<font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">n</font> is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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>  implies  for <font color="Olive" title="b1">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font><br/>  for <font color="Olive" title="b2">n</font> being    <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>   st <font color="Olive" title="b2">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> holds <br/><span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> (<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c3">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font> is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <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> 
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font><br/>  for <font color="Olive" title="b2">n</font> being    <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>   st <font color="Olive" title="b2">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> holds <br/><span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> (<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c3">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font> is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/>

<a NAME="E3:28"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="pre_circ.html#T5" title="PRE_CIRC:th.5">PRE_CIRC:5</a></span>;<br/>
<a NAME="E4:28"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c1">IIG</font> is  <a href="msafree2.html#V1" title="MSAFREE2:attr.1">with_input_V</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="msafree2.html#D4" title="MSAFREE2:def.4">MSAFREE2:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#T2" title="CIRCUIT1:th.2">CIRCUIT1:2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <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>   st <font color="Olive" title="b1">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> holds <br/><span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> (<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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> implies <span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> (<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font> is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E6:28"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c6">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> (<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font> is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">Cn</font> = <span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> (<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font> as   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font> ;<br/>
<a NAME="E8:28"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c4">iv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Cn</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="circuit2.html#T14" target="_self" title="CIRCUIT2:th.14">Th14</a></span>;<br/>
<span class="lab"><font color="Green" title="E30">A7</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="circuit1.html#D2" title="CIRCUIT1:def.2">CIRCUIT1:def 2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="funct_1.html#D10" title="FUNCT_1:def.10">FUNCT_1:def 10</a></span>
;<br/>
<span class="kw">reconsider </span><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="c5">s</font>,<font color="Maroon" title="c3">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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"><span class="kw">now </span></a><div class="add"><span class="kw">thus </span><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="c7">Cn</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#T3" title="CIRCUIT1:th.3">CIRCUIT1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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> &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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> holds <br/><font color="Maroon" title="c7">Cn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="Olive" title="b1">x</font> ) )</span><br/><span class="kw">thus </span><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#T3" title="CIRCUIT1:th.3">CIRCUIT1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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> holds <br/><font color="Maroon" title="c7">Cn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <font color="Maroon" title="c7">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> )</span><br/><span class="kw">assume </span><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">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></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">x9</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"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> (<font color="Maroon" title="c10">x9</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="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="circuit1.html#T17" title="CIRCUIT1:th.17">CIRCUIT1:17</a></span>;<br/><a NAME="E6:28_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">Cn</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x9</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">x9</font>,<font color="Maroon" title="c4">iv</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="circuit2.html#T16" target="_self" title="CIRCUIT2:th.16">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E7:28_2"/>
<font color="Maroon" title="c7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E5:28_2"><span class="lab"><font color="Green" title="E31">A8</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> (<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> (<font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>
<br/>.= 
 <a href="circuit2.html#K7" title="CIRCUIT2:func.7">Following</a> (<font color="Maroon" title="c7">Cn</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="circuit2.html#D9" target="_self" title="CIRCUIT2:def.9">Def9</a></span>
<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="c5">s</font>,<font color="Maroon" title="c3">InpFs</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="funct_4.html#T98" title="FUNCT_4:th.98">FUNCT_4:98</a></span>
;<br/> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="circuit2.html#D6" target="_self" title="CIRCUIT2:def.6">CIRCUIT2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
