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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="petri.html#L1">PT_net_Str</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="boolmark.html#NM1">Boolean_marking</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="petri.html#U2">Transitions</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>case </b></a><a NAME="E1:18_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a>  <a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="boolmark.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><b>case </b></a><a NAME="E1:18_1_1_2"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="boolmark.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E3:18_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="boolmark.html#D5" target="_self">Def5</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a>  <a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_3"><b>case </b></a><a NAME="E1:18_1_1_3"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a>  <a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="boolmark.html#D4" target="_self">Def4</a></i>;<br/><b>thus </b><a NAME="E3:18_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="boolmark.html#E1:18_1_1_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_4"><b>case </b></a><a NAME="E1:18_1_1_4"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E2:18_1_1_4"/><i><font color="Green">E15</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:18_1_1_4"/><i><font color="Green">E16</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E4:18_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/><a NAME="E5:18_1_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:18_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="boolmark.html#K1" target="_self">Bool_marks_of</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:18_1_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E9:18_1_1_4"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E10:18_1_1_4"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E11:18_1_1_4"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18"><i><font color="Green">E13</font></i></a>, <a class="ref" href="boolmark.html#D4" target="_self">Def4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="boolmark.html#K1" target="_self">Bool_marks_of</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:18_1_1_4"/><i><font color="Green">E20</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E14:18_1_1_4"/><i><font color="Green">E21</font></i>: 
 <a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E15:18_1_1_4"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E16:18_1_1_4"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4"><i><font color="Green">E14</font></i></a>, <a class="ref" href="boolmark.html#D5" target="_self">Def5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="boolmark.html#K1" target="_self">Bool_marks_of</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E18:18_1_1_4"/><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E19:18_1_1_4"/>
 <a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E20:18_1_1_4"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E21:18_1_1_4"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4"><i><font color="Green">E14</font></i></a>, <a class="ref" href="boolmark.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E22:18_1_1_4"/><i><font color="Green">E27</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="boolmark.html#E2:18_1_1_4"><i><font color="Green">E15</font></i></a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E23:18_1_1_4"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="boolmark.html#E3:18_1_1_4"><i><font color="Green">E16</font></i></a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E24:18_1_1_4"/><i><font color="Green">E29</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E22:18_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/><a NAME="E25:18_1_1_4"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="txt" href="boolmark.html#E10:18_1_1_4"><i><font color="Green">E18</font></i></a>, <a class="txt" href="boolmark.html#E15:18_1_1_4"><i><font color="Green">E22</font></i></a>, <a class="ref" href="boolmark.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E26:18_1_1_4"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E3:18_1_1_4"><i><font color="Green">E16</font></i></a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E27:18_1_1_4"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E28:18_1_1_4"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:18_1_1_4_1"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E2:18_1_1_4_1"/><b>then </b><i><font color="Green">E34</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>assume </b><a NAME="E3:18_1_1_4_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_4_1_1"/><i><font color="Green">E36</font></i>: 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:18_1_1_4_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:18_1_1_4_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="boolmark.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E4:18_1_1_4_1_1"/><i><font color="Green">E38</font></i>: 
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:18_1_1_4_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> &amp; 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="boolmark.html#E28:18_1_1_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="boolmark.html#E1:18_1_1_4_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:18_1_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E11:18_1_1_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="boolmark.html#E3:18_1_1_4_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="boolmark.html#E1:18_1_1_4_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="boolmark.html#E3:18_1_1_4_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E7:18_1_1_4_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E16:18_1_1_4"><i><font color="Green">E23</font></i></a>, <a class="txt" href="boolmark.html#E2:18_1_1_4_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="boolmark.html#E4:18_1_1_4_1_1"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:18_1_1_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div><a NAME="E30:18_1_1_4"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="boolmark.html#E25:18_1_1_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="boolmark.html#E29:18_1_1_4"><i><font color="Green">E32</font></i></a>);<br/></i><a NAME="E31:18_1_1_4"/><i><font color="Green">E34</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E23:18_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="boolmark.html#E22:18_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> = 
 <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="boolmark.html#E11:18_1_1_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="boolmark.html#E22:18_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="boolmark.html#E28:18_1_1_4"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
 <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="boolmark.html#E13:18_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="boolmark.html#E14:18_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="boolmark.html#E30:18_1_1_4"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="boolmark.html#E20:18_1_1_4"><i><font color="Green">E25</font></i></a>, <a class="txt" href="boolmark.html#E31:18_1_1_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="goboard2.html#T5">GOBOARD2:5</a></i>
;<br/><a NAME="E34:18_1_1_4"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 ;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:18_1_1_4_3"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E2:18_1_1_4_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E3:18_1_1_4_3"/><b>then </b><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:18_1_1_4_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E5:18_1_1_4_3"/><b>then </b><i><font color="Green">E39</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>assume </b><a NAME="E6:18_1_1_4_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_4_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_4_3_1"/><i><font color="Green">E41</font></i>: 
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:18_1_1_4_3_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:18_1_1_4_3_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_3_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="goboard2.html#T5">GOBOARD2:5</a></i>;<br/><a NAME="E4:18_1_1_4_3_1"/><i><font color="Green">E43</font></i>: 
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_3_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:18_1_1_4_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:18_1_1_4_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E7:18_1_1_4_3_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E11:18_1_1_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="boolmark.html#E1:18_1_1_4_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="boolmark.html#E5:18_1_1_4_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="boolmark.html#E6:18_1_1_4_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="boolmark.html#E1:18_1_1_4_3_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="boolmark.html#E3:18_1_1_4_3_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E8:18_1_1_4_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E21:18_1_1_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="boolmark.html#E3:18_1_1_4_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="boolmark.html#E4:18_1_1_4_3_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:18_1_1_4_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div><a NAME="E36:18_1_1_4"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="boolmark.html#E34:18_1_1_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="boolmark.html#E35:18_1_1_4"><i><font color="Green">E36</font></i></a>);<br/></i><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E38:18_1_1_4"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="boolmark.html#E2:18_1_1_4"><i><font color="Green">E15</font></i></a>, <a class="txt" href="boolmark.html#E13:18_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E39:18_1_1_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E13:18_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="boolmark.html#E30:18_1_1_4"><i><font color="Green">E33</font></i></a>, <a class="txt" href="boolmark.html#E38:18_1_1_4"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E40:18_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E31:18_1_1_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="goboard2.html#T5">GOBOARD2:5</a></i>;<br/><a NAME="E41:18_1_1_4"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E11:18_1_1_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="boolmark.html#E13:18_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="boolmark.html#E22:18_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="boolmark.html#E28:18_1_1_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="boolmark.html#E39:18_1_1_4"><i><font color="Green">E39</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_4_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:18_1_1_4_4"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E2:18_1_1_4_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E3:18_1_1_4_4"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:18_1_1_4_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E22:18_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E5:18_1_1_4_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_4"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E6:18_1_1_4_4"/><b>then </b><i><font color="Green">E43</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="boolmark.html#E11:18_1_1_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="boolmark.html#E3:18_1_1_4_4"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E7:18_1_1_4_4"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:18_1_1_4_4"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><i><font color="Green">E45</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="boolmark.html#E8:18_1_1_4_4"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:18_1_1_4_4"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E12:18_1_1_4_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E13:18_1_1_4_4"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E36:18_1_1_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="boolmark.html#E1:18_1_1_4_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="boolmark.html#E11:18_1_1_4_4"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E14:18_1_1_4_4"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E15:18_1_1_4_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E16:18_1_1_4_4"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="boolmark.html#E36:18_1_1_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="boolmark.html#E6:18_1_1_4_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="boolmark.html#E9:18_1_1_4_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="boolmark.html#E13:18_1_1_4_4"><i><font color="Green">E47</font></i></a>, <a class="txt" href="boolmark.html#E14:18_1_1_4_4"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E43:18_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a>  <a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E14:18_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="boolmark.html#E18:18_1_1_4"><i><font color="Green">E24</font></i></a>, <a class="txt" href="boolmark.html#E20:18_1_1_4"><i><font color="Green">E25</font></i></a>, <a class="txt" href="boolmark.html#E41:18_1_1_4"><i><font color="Green">E40</font></i></a>, <a class="ref" href="boolmark.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E44:18_1_1_4"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E9:18_1_1_4"><i><font color="Green">E17</font></i></a>, <a class="txt" href="boolmark.html#E24:18_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="boolmark.html#T8" target="_self">Th8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_4_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:18_1_1_4_5"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E2:18_1_1_4_5"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="boolmark.html#E28:18_1_1_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:18_1_1_4_5"/><b>then </b><i><font color="Green">E43</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="boolmark.html#E11:18_1_1_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="boolmark.html#E1:18_1_1_4_5"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E4:18_1_1_4_5"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_5"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:18_1_1_4_5"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:18_1_1_4_5"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:18_1_1_4_5"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_5"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E9:18_1_1_4_5"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E30:18_1_1_4"><i><font color="Green">E33</font></i></a>, <a class="txt" href="boolmark.html#E1:18_1_1_4_5"><i><font color="Green">E42</font></i></a>, <a class="txt" href="boolmark.html#E8:18_1_1_4_5"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E10:18_1_1_4_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E1:18_1_1_4_5"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E11:18_1_1_4_5"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="boolmark.html#R1" target="_self">is_firable_on</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="boolmark.html#K2" target="_self">Firing</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="boolmark.html#E30:18_1_1_4"><i><font color="Green">E33</font></i></a>, <a class="txt" href="boolmark.html#E3:18_1_1_4_5"><i><font color="Green">E43</font></i></a>, <a class="txt" href="boolmark.html#E6:18_1_1_4_5"><i><font color="Green">E44</font></i></a>, <a class="txt" href="boolmark.html#E9:18_1_1_4_5"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E46:18_1_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="boolmark.html#E13:18_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="boolmark.html#E15:18_1_1_4"><i><font color="Green">E22</font></i></a>, <a class="txt" href="boolmark.html#E44:18_1_1_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="boolmark.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a>  <a href="boolmark.html#K3" target="_self">Firing</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="boolmark.html#R2" target="_self">is_firable_on</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


</div>
