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

<span class="kw">let </span><font color="Maroon" title="c1">PTN</font> be    <a href="petri.html#L1" title="PETRI:struct.1">PT_net_Str</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M0</font> being  <a href="boolmark.html#NM1" title="BOOLMARK:NM.1">Boolean_marking</a> of <font color="Maroon" title="c1">PTN</font><br/> for <font color="Olive" title="b2">Q0</font>, <font color="Olive" title="b3">Q1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="petri.html#U2" title="PETRI:sel.2">Transitions</a> of <font color="Maroon" title="c1">PTN</font>  st <font color="Olive" title="b2">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Olive" title="b1">M0</font> holds <br/>( <font color="Olive" title="b3">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Olive" title="b2">Q0</font>,<font color="Olive" title="b1">M0</font> &amp; <font color="Olive" title="b2">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Olive" title="b1">M0</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M0</font> be   <a href="boolmark.html#NM1" title="BOOLMARK:NM.1">Boolean_marking</a> of <font color="Maroon" title="c1">PTN</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Q0</font>, <font color="Olive" title="b2">Q1</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="petri.html#U2" title="PETRI:sel.2">Transitions</a> of <font color="Maroon" title="c1">PTN</font>  st <font color="Olive" title="b1">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> holds <br/>( <font color="Olive" title="b2">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Olive" title="b1">Q0</font>,<font color="Maroon" title="c2">M0</font> &amp; <font color="Olive" title="b1">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">Q0</font>, <font color="Maroon" title="c4">Q1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="petri.html#U2" title="PETRI:sel.2">Transitions</a> of <font color="Maroon" title="c1">PTN</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> implies ( <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font> &amp; <font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font> &amp; <font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_1_1"/>
( ( <font color="Maroon" title="c3">Q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( <font color="Maroon" title="c3">Q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( <font color="Maroon" title="c3">Q0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( <font color="Maroon" title="c3">Q0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><span class="kw">case </span></a><a NAME="E1:20_1_1_1"/>
( <font color="Maroon" title="c3">Q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font> &amp; <font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_1_1_1"/>
( <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font> &amp; <font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="boolmark.html#D4" target="_self" title="BOOLMARK:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2"><span class="kw">case </span></a><a NAME="E1:20_1_1_2"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
( <font color="Maroon" title="c3">Q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> &amp; <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_1_1_2"/>
<font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="boolmark.html#D4" target="_self" title="BOOLMARK:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font></span><br/><a NAME="E3:20_1_1_2"/>
<font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_2"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/><span class="kw">hence </span><a NAME="E4:20_1_1_2"/>
<font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E1:20_1_1_2"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="boolmark.html#D5" target="_self" title="BOOLMARK:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_3"><span class="kw">case </span></a><a NAME="E1:20_1_1_3"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
( <font color="Maroon" title="c3">Q0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font> &amp; <font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_1_1_3"/>
<font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="boolmark.html#D4" target="_self" title="BOOLMARK:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font></span><br/><span class="kw">thus </span><a NAME="E3:20_1_1_3"/>
<font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E1:20_1_1_3"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_4"><span class="kw">case </span></a><a NAME="E1:20_1_1_4"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
( <font color="Maroon" title="c3">Q0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">Q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font> &amp; <font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> )</span><br/><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font> &amp; <font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> )</span><br/><a NAME="E2:20_1_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E3:20_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:20_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T153" title="SEQ_4:th.153">SEQ_4:153</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">m</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">M4</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="boolmark.html#K1" title="BOOLMARK:func.1">Bool_marks_of</a> <font color="Maroon" title="c1">PTN</font><span class="kw"> such that </span><br/><a NAME="E7:20_1_1_4"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M4</font>
 <span class="kw">and </span><br/><a NAME="E8:20_1_1_4"/>
 <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c4">Q1</font>,<span class="p1">(<span class="default"><a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">M4</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:20_1_1_4"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:20_1_1_4"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="boolmark.html#D5" target="_self" title="BOOLMARK:def.5">Def5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">M3</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="boolmark.html#K1" title="BOOLMARK:func.1">Bool_marks_of</a> <font color="Maroon" title="c1">PTN</font><span class="kw"> such that </span><br/><a NAME="E12:20_1_1_4"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M3</font>
 <span class="kw">and </span><br/><a NAME="E13:20_1_1_4"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
 <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M3</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:20_1_1_4"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c2">M0</font>
 <span class="kw">and </span><br/><a NAME="E15:20_1_1_4"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="boolmark.html#D5" target="_self" title="BOOLMARK:def.5">Def5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E17:20_1_1_4"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E12:20_1_1_4"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c9">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">M</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="boolmark.html#K1" title="BOOLMARK:func.1">Bool_marks_of</a> <font color="Maroon" title="c1">PTN</font><span class="kw"> such that </span><br/><a NAME="E20:20_1_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">M</font>
 <span class="kw">and </span><br/><a NAME="E21:20_1_1_4"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Maroon" title="c2">M0</font>
 <span class="kw">and </span><br/><a NAME="E22:20_1_1_4"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c2">M0</font>
 <span class="kw">and </span><br/><a NAME="E23:20_1_1_4"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E1:20_1_1_4"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="boolmark.html#D4" target="_self" title="BOOLMARK:def.4">Def4</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font> implies <font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> );<br/><a NAME="E24:20_1_1_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E25:20_1_1_4"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E26:20_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A18</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><div><span class="lab"><font color="Green" title="E29">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><a NAME="E1:20_1_1_4_1"/><span class="lab"><font color="Green" title="E30">A20</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><span class="kw">assume </span><a NAME="E2:20_1_1_4_1"/><span class="lab"><font color="Green" title="E31">A21</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_1_1_4_1_1"/><span class="lab"><font color="Green" title="E32">A22</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/><a NAME="E2:20_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A23</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="boolmark.html#T8" target="_self" title="BOOLMARK:th.8">Th8</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:20_1_1_4_1_1"/><span class="lab"><font color="Green" title="E34">A24</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4_1_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:20_1_1_4_1_1"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:20_1_1_4"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:20_1_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20_1_1_4"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E1:20_1_1_4_1"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E2:20_1_1_4_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E1:20_1_1_4_1_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="txt" href="#E2:20_1_1_4_1_1"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E6:20_1_1_4_1_1"/>
<font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:20_1_1_4"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E1:20_1_1_4_1"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E3:20_1_1_4_1_1"><span class="lab"><font color="Green" title="E34">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:20_1_1_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E28:20_1_1_4"/><span class="lab"><font color="Green" title="E30">A25</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:20_1_1_4"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="#E14:20_1_1_4"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="boolmark.html#T8" target="_self" title="BOOLMARK:th.8">Th8</a></span>;<br/><a NAME="E29:20_1_1_4"/><span class="lab"><font color="Green" title="E31">A26</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E28:20_1_1_4"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="txt" href="#E27:20_1_1_4"><span class="lab"><font color="Green" title="E29">A19</font></span></a>);<br/></span><div><span class="lab"><font color="Green" title="E32">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_1_1_4_2"/><span class="lab"><font color="Green" title="E33">A28</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font>
 <span class="kw">and </span><br/><a NAME="E2:20_1_1_4_2"/><span class="lab"><font color="Green" title="E34">A29</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c13">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:20_1_1_4_2"/><span class="lab"><font color="Green" title="E35">A30</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1_4_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><a NAME="E5:20_1_1_4_2"/>
( <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:20_1_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">Q0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:20_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A31</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">j</font> = <font color="Maroon" title="c13">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E9:20_1_1_4_2"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1_1_4_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a></span>;<br/><a NAME="E10:20_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A32</font></span>: 
<font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:20_1_1_4"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="txt" href="#E1:20_1_1_4_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a></span>;<br/><a NAME="E11:20_1_1_4_2"/><span class="lab"><font color="Green" title="E38">A33</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E12:20_1_1_4_2"/><span class="lab"><font color="Green" title="E39">A34</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:20_1_1_4"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="txt" href="#E1:20_1_1_4_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:20_1_1_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20_1_1_4"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E2:20_1_1_4_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:20_1_1_4_2"/>
( <font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20_1_1_4"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E29:20_1_1_4"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="txt" href="#E2:20_1_1_4_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="txt" href="#E12:20_1_1_4_2"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="txt" href="#E7:20_1_1_4_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E10:20_1_1_4_2"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="txt" href="#E11:20_1_1_4_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font> implies <font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> );<br/><div><span class="lab"><font color="Green" title="E33">A35</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:20_1_1_4_3"/><span class="lab"><font color="Green" title="E34">A36</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><a NAME="E2:20_1_1_4_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E3:20_1_1_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A37</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><a NAME="E4:20_1_1_4_3"/><span class="lab"><font color="Green" title="E36">A38</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:20_1_1_4_3_1"/><span class="lab"><font color="Green" title="E37">A39</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/><a NAME="E2:20_1_1_4_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A40</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="seq_4.html#T153" title="SEQ_4:th.153">SEQ_4:153</a></span>;<br/><a NAME="E3:20_1_1_4_3_1"/><span class="lab"><font color="Green" title="E39">A41</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4_3_1"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:20_1_1_4_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E5:20_1_1_4_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E6:20_1_1_4_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20_1_1_4"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E3:20_1_1_4_3"><span class="lab"><font color="Green" title="E35">A37</font></span></a>, <a class="txt" href="#E1:20_1_1_4_3"><span class="lab"><font color="Green" title="E34">A36</font></span></a>, <a class="txt" href="#E1:20_1_1_4_3_1"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="txt" href="#E2:20_1_1_4_3_1"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E7:20_1_1_4_3_1"/>
<font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_1_1_4"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E4:20_1_1_4_3"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="txt" href="#E3:20_1_1_4_3_1"><span class="lab"><font color="Green" title="E39">A41</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:20_1_1_4_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E32:20_1_1_4"/><span class="lab"><font color="Green" title="E34">A42</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
 <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20_1_1_4"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E26:20_1_1_4"><span class="lab"><font color="Green" title="E28">A18</font></span></a></span>
<br/>.= 
 <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_1_1_4"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E13:20_1_1_4"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E29:20_1_1_4"><span class="lab"><font color="Green" title="E31">A26</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20_1_1_4"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E3:20_1_1_4"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="seq_4.html#T153" title="SEQ_4:th.153">SEQ_4:153</a></span>
;<br/><a NAME="E34:20_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A43</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><a NAME="E35:20_1_1_4"/><span class="lab"><font color="Green" title="E36">A44</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E34:20_1_1_4"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="txt" href="#E31:20_1_1_4"><span class="lab"><font color="Green" title="E33">A35</font></span></a>);<br/></span><div><span class="lab"><font color="Green" title="E37">A45</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:20_1_1_4_5"/><span class="lab"><font color="Green" title="E38">A46</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font>
 <span class="kw">and </span><br/><a NAME="E2:20_1_1_4_5"/><span class="lab"><font color="Green" title="E39">A47</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c13">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:20_1_1_4_5"/><span class="lab"><font color="Green" title="E40">A48</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1_4_5"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">j</font> = <font color="Maroon" title="c13">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E6:20_1_1_4_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E7:20_1_1_4_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A49</font></span>: 
<font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:20_1_1_4"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E1:20_1_1_4_5"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E4:20_1_1_4_5"><span class="lab"><font color="Green" title="E40">A48</font></span></a></span>;<br/><a NAME="E8:20_1_1_4_5"/>
( <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4_5"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E9:20_1_1_4_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A50</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">Q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="lab"><font color="Green" title="E43">A51</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20_1_1_4_5"><span class="lab"><font color="Green" title="E42">A50</font></span></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>
;<br/><a NAME="E11:20_1_1_4_5"/><span class="lab"><font color="Green" title="E44">A52</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4_5"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E12:20_1_1_4_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E13:20_1_1_4_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A53</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4_5"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E14:20_1_1_4_5"/><span class="lab"><font color="Green" title="E46">A54</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_4"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E15:20_1_1_4_5"/><span class="kw">then </span>
<font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">Q0</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">Q1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20_1_1_4"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E2:20_1_1_4_5"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E13:20_1_1_4_5"><span class="lab"><font color="Green" title="E45">A53</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:20_1_1_4_5"/>
( <font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">M4</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">i</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20_1_1_4"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E35:20_1_1_4"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E2:20_1_1_4_5"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E13:20_1_1_4_5"><span class="lab"><font color="Green" title="E45">A53</font></span></a>, <a class="txt" href="#E14:20_1_1_4_5"><span class="lab"><font color="Green" title="E46">A54</font></span></a>, <a class="txt" href="#E10:20_1_1_4_5"><span class="lab"><font color="Green" title="E43">A51</font></span></a>, <a class="txt" href="#E7:20_1_1_4_5"><span class="lab"><font color="Green" title="E41">A49</font></span></a>, <a class="txt" href="#E11:20_1_1_4_5"><span class="lab"><font color="Green" title="E44">A52</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E37:20_1_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:20_1_1_4"><span class="lab"><font color="Green" title="E24">A14</font></span></a></span>;<br/><a NAME="E38:20_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">M</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_1_1_4"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E29:20_1_1_4"><span class="lab"><font color="Green" title="E31">A26</font></span></a></span>;<br/><a NAME="E39:20_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">Q1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Maroon" title="c8">M3</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:20_1_1_4"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="txt" href="#E12:20_1_1_4"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E32:20_1_1_4"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="txt" href="#E26:20_1_1_4"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="txt" href="#E4:20_1_1_4"><span class="lab"><font color="Green" title="E16">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E40:20_1_1_4"/>
<font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20_1_1_4"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E7:20_1_1_4"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E9:20_1_1_4"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E36:20_1_1_4"><span class="lab"><font color="Green" title="E37">A45</font></span></a>, <a class="ref" href="boolmark.html#D4" target="_self" title="BOOLMARK:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font></span><br/><a NAME="E41:20_1_1_4"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">Q0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:20_1_1_4"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E42:20_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">Q0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Maroon" title="c2">M0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20_1_1_4"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="boolmark.html#T8" target="_self" title="BOOLMARK:th.8">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E43:20_1_1_4"/>
<font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:20_1_1_4"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E14:20_1_1_4"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E30:20_1_1_4"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="ref" href="boolmark.html#D4" target="_self" title="BOOLMARK:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:20"/>
( <font color="Maroon" title="c4">Q1</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a>  <a href="boolmark.html#K3" title="BOOLMARK:func.3">Firing</a> <font color="Maroon" title="c3">Q0</font>,<font color="Maroon" title="c2">M0</font> &amp; <font color="Maroon" title="c3">Q0</font> <a href="boolmark.html#R2" title="BOOLMARK:pred.2">is_firable_on</a> <font color="Maroon" title="c2">M0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
