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<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> st <br/>(  <a href="circuit1.html#K4">size</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K4">size</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>4</sub></font> ) &amp; ex <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K4">size</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>4</sub></font> ) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_2_1"/><i><font color="Green">E11</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:11_1_2_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>]
 <b>and </b><br/><a NAME="E4:11_1_2_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="circuit2.html#E1:11_1_2_1"><i><font color="Green">E11</font></i></a>);<br/></i><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E6:11_1_2_1"/><i><font color="Green">E14</font></i>: 
 <a href="circuit1.html#K4">size</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E7:11_1_2_1"/><i><font color="Green">E15</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K4">size</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E8:11_1_2_1"/><i><font color="Green">E16</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K4">size</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E9:11_1_2_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E3:11_1_2_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E11:11_1_2_1"/><i><font color="Green">E18</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K4">size</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E12:11_1_2_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E7:11_1_2_1"><i><font color="Green">E15</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E14:11_1_2_1"/><i><font color="Green">E20</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K4">size</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E15:11_1_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E8:11_1_2_1"><i><font color="Green">E16</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_2_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font> or ( not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:11_1_2_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E3:11_1_2_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="msafree2.html#T12">MSAFREE2:12</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E5:11_1_2_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:11_1_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msafree2.html#T12">MSAFREE2:12</a></i>;<br/><font color="Maroon">c<sub>9</sub></font> = 
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="circuit2.html#E12:11_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="circuit2.html#E1:11_1_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="circuit2.html#E3:11_1_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="circuit2.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="circuit2.html#E15:11_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="circuit2.html#E1:11_1_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="circuit2.html#E5:11_1_2_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="circuit2.html#T3" target="_self">Th3</a></i>
;<br/><b>hence </b><a NAME="E7:11_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="circuit2.html#E9:11_1_2_1"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:11_1_2_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>9</sub></font> = 
 <a href="trees_4.html#K1">root-tree</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="circuit2.html#E12:11_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="circuit2.html#T5" target="_self">Th5</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="circuit2.html#E15:11_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="circuit2.html#E1:11_1_2_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="circuit2.html#T5" target="_self">Th5</a></i>
;<br/><b>hence </b><a NAME="E3:11_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="circuit2.html#E9:11_1_2_1"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:11_1_2_1_1_3"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:11_1_2_1_1_3"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E3:11_1_2_1_1_3"/>
<span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E4:11_1_2_1_1_3"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:11_1_2_1_1_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:11_1_2_1_1_3"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E2:11_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E7:11_1_2_1_1_3"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E11:11_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="circuit1.html#T13">CIRCUIT1:13</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:11_1_2_1_1_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E14:11_1_2_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="circuit2.html#E5:11_1_2_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="circuit1.html#T13">CIRCUIT1:13</a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>;<br/><a NAME="E10:11_1_2_1_1_3"/><i><font color="Green">E28</font></i>: 
 <a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font>, <a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="circuit2.html#D2" target="_self">Def2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E12:11_1_2_1_1_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="pralg_1.html#K15">..</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E13:11_1_2_1_1_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E12:11_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="circuit2.html#E4:11_1_2_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="circuit2.html#E7:11_1_2_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="circuit2.html#T1" target="_self">Th1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="trees_3.html#V6">DTree-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E15:11_1_2_1_1_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="pralg_1.html#K15">..</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E16:11_1_2_1_1_3"/><i><font color="Green">E32</font></i>: 
<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"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E15:11_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="circuit2.html#E4:11_1_2_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="circuit2.html#E9:11_1_2_1_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="circuit2.html#E10:11_1_2_1_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="circuit2.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E17:11_1_2_1_1_3"/><i><font color="Green">E33</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="circuit2.html#E12:11_1_2_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="circuit2.html#E15:11_1_2_1_1_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/><a NAME="E18:11_1_2_1_1_3"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E19:11_1_2_1_1_3"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E20:11_1_2_1_1_3"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E21:11_1_2_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E17:11_1_2_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E23:11_1_2_1_1_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E24:11_1_2_1_1_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E9:11_1_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="circuit2.html#E13:11_1_2_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="circuit2.html#E16:11_1_2_1_1_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/><a NAME="E25:11_1_2_1_1_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E23:11_1_2_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E26:11_1_2_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E17:11_1_2_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circuit2.html#E20:11_1_2_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="circuit2.html#E23:11_1_2_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E28:11_1_2_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="circuit2.html#E13:11_1_2_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="circuit2.html#E16:11_1_2_1_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E29:11_1_2_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="circuit2.html#E4:11_1_2_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <b>by </b><i><a class="txt" href="circuit2.html#E17:11_1_2_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circuit2.html#E20:11_1_2_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="circuit2.html#E25:11_1_2_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msafree2.html#T14">MSAFREE2:14</a></i>;<br/><a NAME="E31:11_1_2_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="circuit2.html#E7:11_1_2_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="circuit2.html#E9:11_1_2_1_1_3"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E32:11_1_2_1_1_3"/><b>then </b><i><font color="Green">E38</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="trees_4.html#K4">-tree</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="circuit2.html#E4:11_1_2_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <b>by </b><i><a class="txt" href="circuit2.html#E17:11_1_2_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circuit2.html#E20:11_1_2_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="circuit2.html#E25:11_1_2_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="msafree2.html#T14">MSAFREE2:14</a></i>;<br/><a NAME="E34:11_1_2_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E32:11_1_2_1_1_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="msafree2.html#T13">MSAFREE2:13</a></i>;<br/><a NAME="E35:11_1_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E17:11_1_2_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circuit2.html#E20:11_1_2_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="circuit2.html#E25:11_1_2_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E36:11_1_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E37:11_1_2_1_1_3"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K4">size</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E11:11_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="circuit2.html#E5:11_1_2_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="circuit2.html#E7:11_1_2_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="circuit1.html#T12">CIRCUIT1:12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E39:11_1_2_1_1_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E17:11_1_2_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circuit2.html#E25:11_1_2_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E40:11_1_2_1_1_3"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E12:11_1_2_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="circuit2.html#E17:11_1_2_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circuit2.html#E25:11_1_2_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/><a NAME="E41:11_1_2_1_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E32:11_1_2_1_1_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="msafree2.html#T13">MSAFREE2:13</a></i>;<br/><a NAME="E42:11_1_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E17:11_1_2_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circuit2.html#E20:11_1_2_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="circuit2.html#E25:11_1_2_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E43:11_1_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E44:11_1_2_1_1_3"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K4">size</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E14:11_1_2_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="circuit2.html#E5:11_1_2_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="circuit2.html#E9:11_1_2_1_1_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="circuit1.html#T12">CIRCUIT1:12</a></i>;<br/><a NAME="E45:11_1_2_1_1_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E15:11_1_2_1_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="circuit2.html#E17:11_1_2_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circuit2.html#E25:11_1_2_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="circuit2.html#E39:11_1_2_1_1_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/><a NAME="E46:11_1_2_1_1_3"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="circuit1.html#K4">size</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E4:11_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="circuit2.html#E24:11_1_2_1_1_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="circuit2.html#E37:11_1_2_1_1_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="circuit2.html#E40:11_1_2_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="circuit2.html#E44:11_1_2_1_1_3"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E47:11_1_2_1_1_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="circuit2.html#E17:11_1_2_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="circuit2.html#E20:11_1_2_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="circuit2.html#E25:11_1_2_1_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E48:11_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="circuit2.html#E6:11_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="circuit2.html#E4:11_1_2_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="circuit2.html#E46:11_1_2_1_1_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="circuit1.html#T15">CIRCUIT1:15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:11_1_2"/>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K4">size</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="circuit1.html#K4">size</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K3" target="_self">Fix_inp_ext</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>


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