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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="circcomb.html#M1">FinSeqLen</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="circcomb.html#K7">1GateCircStr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K9">1GateCircuit</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/>
<a NAME="E1:3"/><i><font color="Green">E2</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/>
<a NAME="E2:3"/><i><font color="Green">E3</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="circcomb.html#D6">CIRCCOMB:def 6</a></i>;<br/>
<a NAME="E3:3"/><i><font color="Green">E4</font></i>: 
(  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="circcomb.html#T49">CIRCCOMB:49</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><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="E1:3_1"/>
<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 <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<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 <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> ;<br/><a NAME="E3:3_1"/><i><font color="Green">E5</font></i>: 
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:3_1_1"/><i><font color="Green">E6</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/>
<a NAME="E2:3_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="circcmb2.html#E2:3"><i><font color="Green">E3</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:3_1_1"/><b>then </b><i><font color="Green">E7</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="circcmb2.html#E1:3"><i><font color="Green">E2</font></i></a>, <a class="txt" href="circcmb2.html#E1:3_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:3_1_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:3_1_1_1"/><b>then </b><i><font color="Green">E9</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="circcmb2.html#E3:3_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E3:3_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="circcmb2.html#E1:3_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="circcmb2.html#E3:3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="circcmb2.html#E2:3_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="circuit2.html#D5">CIRCUIT2:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:3_1_1"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="circcmb2.html#E3:3_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:3_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="circcmb2.html#E2:3"><i><font color="Green">E3</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> implies  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="circcomb.html#K9">1GateCircuit</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="circuit1.html#K3">depends_on_in</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="circcomb.html#K9">1GateCircuit</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="circuit1.html#K3">depends_on_in</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; (  <a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> implies ( <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="circuit1.html#K3">depends_on_in</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="circuit1.html#K3">depends_on_in</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> ) ) ) )
 <b>by </b><i><a class="txt" href="circcmb2.html#E3:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="circcmb2.html#T1" target="_self">Th1</a>, <a class="ref" href="circcomb.html#T48">CIRCCOMB:48</a>, <a class="ref" href="circuit2.html#D5">CIRCUIT2:def 5</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:3_1"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="circcmb2.html#E3:3_1"><i><font color="Green">E5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:3"/>
 <a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circcmb2.html#E1:3"><i><font color="Green">E2</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circuit2.html#D6">CIRCUIT2:def 6</a><br/>


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