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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="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:()"><i><font color="Red">::  thesis: </font></i></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">iv</font> being   <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Olive" title="b1">SCS</font>  st <font color="Olive" title="b4">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b2">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>  holds <br/>for <font color="Olive" title="b5">k</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> <br/> for <font color="Olive" title="b6">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>  st  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Olive" title="b6">v</font>,<font color="Olive" title="b1">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">k</font> holds <br/><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="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="b5">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b6">v</font>,<font color="Olive" title="b4">iv</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">iv</font> being   <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</font>  st <font color="Olive" title="b3">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b1">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>  holds <br/>for <font color="Olive" title="b4">k</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> <br/> for <font color="Olive" title="b5">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>  st  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Olive" title="b5">v</font>,<font color="Maroon" title="c2">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">k</font> holds <br/><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="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="b4">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b5">v</font>,<font color="Olive" title="b3">iv</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">iv</font> being   <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</font>  st <font color="Olive" title="b2">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>for <font color="Olive" title="b3">k</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> <br/> for <font color="Olive" title="b4">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>  st  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Olive" title="b4">v</font>,<font color="Maroon" title="c2">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> holds <br/><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="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="b3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b4">v</font>,<font color="Olive" title="b2">iv</font> )</span><br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E23">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">iv</font> being   <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</font>  st <font color="Olive" title="b2">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>for <font color="Olive" title="b3">k</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> <br/> for <font color="Olive" title="b4">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>  st  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Olive" title="b4">v</font>,<font color="Maroon" title="c2">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> holds <br/><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="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="b3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b4">v</font>,<font color="Olive" title="b2">iv</font></span><br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green" title="E24">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">iv</font> being   <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</font>  st <font color="Olive" title="b1">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>for <font color="Olive" title="b2">k</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> <br/> for <font color="Olive" title="b3">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>  st  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Olive" title="b3">v</font>,<font color="Maroon" title="c2">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/><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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b1">iv</font></span><br/><b>let </b><font color="Maroon" title="c5">iv</font> be    <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">k</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> <br/> for <font color="Olive" title="b2">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>  st  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Olive" title="b2">v</font>,<font color="Maroon" title="c2">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/><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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b2">v</font>,<font color="Maroon" title="c5">iv</font> )</span><br/>



<b>assume </b><a NAME="E3:26"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c5">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</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> <br/> for <font color="Olive" title="b2">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>  st  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Olive" title="b2">v</font>,<font color="Maroon" title="c2">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/><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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b2">v</font>,<font color="Maroon" title="c5">iv</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>for <font color="Olive" title="b1">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>  st  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Olive" title="b1">v</font>,<font color="Maroon" title="c2">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/><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="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b1">v</font>,<font color="Maroon" title="c5">iv</font>;<br/>
<a NAME="E4:26"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c2">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</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="c6">v</font>,<font color="Maroon" title="c5">iv</font> )</span><br/>

<b>assume </b><a NAME="E1:26_1"/>
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c2">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</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="c6">v</font>,<font color="Maroon" title="c5">iv</font></span><br/>

<a NAME="E2:26_1"/><b>then </b><i><font color="Green" title="E27">A5</font></i>: 
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c2">SCS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/>
<i><font color="Green" title="E28">A6</font></i>: <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
 <a href="circuit2.html#K8" title="CIRCUIT2:func.8">InitialComp</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">InpFs</font>
<b>by </b><i><a class="ref" href="circuit2.html#D9" target="_self" title="CIRCUIT2:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="circuit2.html#K1" title="CIRCUIT2:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="circuit1.html#K2" title="CIRCUIT1:func.2">-th_InputValues</a> <font color="Maroon" title="c3">InpFs</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="circuit1.html#K1" title="CIRCUIT1:func.1">Set-Constants</a> <font color="Maroon" title="c2">SCS</font></span>)</span>

;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_2"/>
( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> or <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K1" title="MSAFREE2:func.1">SortsWithConstants</a> <font color="Maroon" title="c1">IIG</font> )
 </span><b>by </b><i><a class="txt" href="#E2:26_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="circuit1.html#T19" title="CIRCUIT1:th.19">CIRCUIT1:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_1"><b>suppose </b></a><a NAME="E1:26_1_2_1"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</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="c6">v</font>,<font color="Maroon" title="c5">iv</font></span><br/><div class="add"><a NAME="E2:26_1_2_1"/>
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="msafree2.html#K1" title="MSAFREE2:func.1">SortsWithConstants</a> <font color="Maroon" title="c1">IIG</font>
 <b>by </b><i><a class="ref" href="msafree2.html#T6" title="MSAFREE2:th.6">MSAFREE2:6</a></i>;<br/><a NAME="E3:26_1_2_1"/><b>then </b>
not <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K1" title="MSAFREE2:func.1">SortsWithConstants</a> <font color="Maroon" title="c1">IIG</font>
 <b>by </b><i><a class="txt" href="#E1:26_1_2_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:26_1_2_1"/><b>then </b><i><font color="Green" title="E30">A8</font></i>: 
not <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="circuit1.html#K1" title="CIRCUIT1:func.1">Set-Constants</a> <font color="Maroon" title="c2">SCS</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E5:26_1_2_1"/><i><font color="Green" title="E31">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="circuit1.html#K2" title="CIRCUIT1:func.2">-th_InputValues</a> <font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>thus </b><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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="circuit2.html#K1" title="CIRCUIT2:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="circuit1.html#K2" title="CIRCUIT1:func.2">-th_InputValues</a> <font color="Maroon" title="c3">InpFs</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="#E3:26_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="#E4:26_1_2_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="circuit1.html#K2" title="CIRCUIT1:func.2">-th_InputValues</a> <font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="#E1:26_1_2_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="#E5:26_1_2_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c5">iv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="#E2:26"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E3:26"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="circuit1.html#D2" title="CIRCUIT1:def.2">CIRCUIT1:def 2</a></i>
<br/>.= 
 <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">iv</font>
<b>by </b><i><a class="txt" href="#E1:26_1_2_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="circuit2.html#T10" target="_self" title="CIRCUIT2:th.10">Th10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_2"><b>suppose </b></a><a NAME="E1:26_1_2_2"/><i><font color="Green" title="E29">A10</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K1" title="MSAFREE2:func.1">SortsWithConstants</a> <font color="Maroon" title="c1">IIG</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</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="c6">v</font>,<font color="Maroon" title="c5">iv</font></span><br/><div class="add"><a NAME="E2:26_1_2_2"/><b>then </b>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="circuit1.html#K1" title="CIRCUIT1:func.1">Set-Constants</a> <font color="Maroon" title="c2">SCS</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>hence </b><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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><a href="circuit1.html#K1" title="CIRCUIT1:func.1">Set-Constants</a> <font color="Maroon" title="c2">SCS</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="#E3:26_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">iv</font>
<b>by </b><i><a class="txt" href="#E1:26_1_2_2"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="circuit2.html#T11" target="_self" title="CIRCUIT2:th.11">Th11</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:26"/><i><font color="Green" title="E27">A11</font></i>: 
for <font color="Olive" title="b1">k</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>reconsider </b><font color="Maroon" title="c7">Ck</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">k</font> as   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font> ;<br/>
<b>assume </b><a NAME="E2:26_2"/><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E3:26_2"/><i><font color="Green" title="E29">A13</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="#E1:26"><i><font color="Green" title="E23">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="E30">A14</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</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">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E2:26"><i><font color="Green" title="E24">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="#E1:26"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E3:26_2"><i><font color="Green" title="E29">A13</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="E5:26_2"/><i><font color="Green" title="E31">A15</font></i>: 
<font color="Maroon" title="c5">iv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ck</font>
 
<b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E3:26"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="circuit2.html#T14" target="_self" title="CIRCUIT2:th.14">Th14</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">v</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c8">v</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">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">v</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="c8">v</font>,<font color="Maroon" title="c5">iv</font> )</span><br/>

<b>assume </b><a NAME="E6:26_2"/><i><font color="Green" title="E32">A16</font></i>: 
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c8">v</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">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">v</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="c8">v</font>,<font color="Maroon" title="c5">iv</font></span><br/>

<b>thus </b><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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">v</font> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K7" title="CIRCUIT2:func.7">Following</a> <font color="Maroon" title="c7">Ck</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</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>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">v</font>
<b>by </b><i><a class="ref" href="circuit2.html#D9" target="_self" title="CIRCUIT2:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c7">Ck</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">v</font>
<b>by </b><i><a class="txt" href="#E3:26"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="#E4:26_2"><i><font color="Green" title="E30">A14</font></i></a>, <a class="txt" href="#E5:26_2"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="funct_4.html#T79" title="FUNCT_4:th.79">FUNCT_4:79</a></i>
<br/>.= 
 <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c5">iv</font>
<b>by </b><i><a class="txt" href="#E2:26_2"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="#E6:26_2"><i><font color="Green" title="E32">A16</font></i></a>, <a class="ref" href="circuit2.html#T13" target="_self" title="CIRCUIT2:th.13">Th13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:26"/>
for <font color="Olive" title="b1">k</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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="#E4:26"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="#E5:26"><i><font color="Green" title="E27">A11</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
