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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a> <a href="msafree2.html#V5">monotonic</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="circuit1.html#NM2">InputFuncs</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E24</font></i>: 
(  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V3">constant</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="msafree2.html#V1">with_input_V</a>
 
<b>by </b><i><a class="ref" href="msafree2.html#D4">MSAFREE2:def 4</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E3:28"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K7">depth</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:28"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="circuit2.html#E1:28"><i><font color="Green">E24</font></i></a>, <a class="ref" href="pre_circ.html#T8">PRE_CIRC:8</a></i>;<br/>
<i><font color="Green">E28</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="circuit1.html#K2">-th_InputValues</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="circuit2.html#E2:28"><i><font color="Green">E25</font></i></a>, <a class="ref" href="circuit1.html#D2">CIRCUIT1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i><a class="txt" href="circuit2.html#E1:28"><i><font color="Green">E24</font></i></a>, <a class="txt" href="circuit2.html#E4:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="circuit2.html#K9" target="_self">Computation</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0 as    <a href="msafree2.html#M1">InputValues</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="circuit2.html#E2:28"><i><font color="Green">E25</font></i></a>, <a class="ref" href="circuit1.html#T2">CIRCUIT1:2</a></i>;<br/>
<a NAME="E8:28"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="circuit2.html#E1:28"><i><font color="Green">E24</font></i></a>, <a class="ref" href="circuit2.html#T14" target="_self">Th14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="circuit2.html#K9" target="_self">Computation</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>2</sub></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">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/><b>thus </b><a NAME="E2:28_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:28_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E5:28_2"/><i><font color="Green">E30</font></i>: 
 <a href="circuit1.html#K6">depth</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E3:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="circuit1.html#T18">CIRCUIT1:18</a></i>;<br/><a NAME="E6:28_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K5" target="_self">IGValue</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:28"><i><font color="Green">E24</font></i></a>, <a class="ref" href="circuit2.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E7:28_2"/>
 <a href="circuit1.html#K6">depth</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="circuit2.html#E5:28_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E8:28_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:28"><i><font color="Green">E24</font></i></a>, <a class="txt" href="circuit2.html#E6:28_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="circuit2.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><span class="p1">(<span class="default"><a href="circuit2.html#K9" target="_self">Computation</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K9" target="_self">Computation</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
<br/>.= 
 <a href="circuit2.html#K7" target="_self">Following</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="circuit1.html#K2">-th_InputValues</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="circuit2.html#D9" target="_self">Def9</a></i>
<br/>.= 
 <a href="circuit2.html#K6" target="_self">Following</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="circuit2.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="circuit2.html#E5:28"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
 <a href="circuit2.html#K6" target="_self">Following</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" target="_self">Computation</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="circuit2.html#E8:28"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_4.html#T79">FUNCT_4:79</a></i>
;<br/> <i><font color="Red">:: according to </font></i><a class="ref" href="circuit2.html#D6" target="_self">CIRCUIT2:def 6</a><br/>


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