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<span class="kw">let </span><font color="Maroon" title="c1">P</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">N</font> being    <a href="pnproc_1.html#M2" title="PNPROC_1:mode.2">Petri_net</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b2">C1</font>, <font color="Olive" title="b3">C2</font> being   <a href="pnproc_1.html#NM5" title="PNPROC_1:NM.5">firing-sequence</a> of <font color="Olive" title="b1">N</font> holds   <a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Olive" title="b3">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <font color="Olive" title="b3">C2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <font color="Olive" title="b2">C1</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">N</font> be    <a href="pnproc_1.html#M2" title="PNPROC_1:mode.2">Petri_net</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font> being   <a href="pnproc_1.html#NM5" title="PNPROC_1:NM.5">firing-sequence</a> of <font color="Maroon" title="c2">N</font> holds   <a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Olive" title="b2">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <font color="Olive" title="b2">C2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <font color="Olive" title="b1">C1</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">C1</font>, <font color="Maroon" title="c4">C2</font> be   <a href="pnproc_1.html#NM5" title="PNPROC_1:NM.5">firing-sequence</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <font color="Maroon" title="c3">C1</font></span>)</span></span><br/>







<span class="kw">consider </span><font color="Maroon" title="c5">F</font> being   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:61"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 <a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<font color="Maroon" title="c5">F</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">P</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>)</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E3:61"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:61"/><span class="lab"><font color="Green" title="E43">A3</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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span> holds <br/><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span> <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="ref" href="pnproc_1.html#D10" target="_self" title="PNPROC_1:def.10">Def11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">F1</font> being   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E6:61"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
 <a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<font color="Maroon" title="c6">F1</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">P</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>)</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E7:61"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F1</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="c3">C1</font>
 <span class="kw">and </span><br/><a NAME="E8:61"/><span class="lab"><font color="Green" title="E46">A6</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="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">C1</font> holds <br/><font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</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="ref" href="pnproc_1.html#D10" target="_self" title="PNPROC_1:def.10">Def11</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">F2</font> being   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E10:61"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
 <a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <font color="Maroon" title="c4">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K4" title="FUNCT_7:func.4">compose</a> (<font color="Maroon" title="c7">F2</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">P</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>)</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E11:61"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</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="c4">C2</font>
 <span class="kw">and </span><br/><a NAME="E12:61"/><span class="lab"><font color="Green" title="E49">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="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">C2</font> holds <br/><font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C2</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="ref" href="pnproc_1.html#D10" target="_self" title="PNPROC_1:def.10">Def11</a></span>;<br/>
<a NAME="E13:61"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <font color="Maroon" title="c3">C1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">P</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T37" target="_self" title="PNPROC_1:th.37">Th40</a></span>;<br/>
<a NAME="E14:61"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</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">C1</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">C2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E15:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A11</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="c6">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E11:61"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E16:61"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <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="c6">F1</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <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="c6">F1</font> implies <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1"/><span class="lab"><font color="Green" title="E53">A13</font></span>: 
<font color="Maroon" title="c8">k</font> <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="c6">F1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/>

<a NAME="E2:61_1"/><span class="lab"><font color="Green" title="E54">A14</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:61_1"/><span class="lab"><font color="Green" title="E55">A15</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">C1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:61_1"/><span class="lab"><font color="Green" title="E56">A16</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E2:61_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:61_1"/><span class="lab"><font color="Green" title="E57">A17</font></span>: 
<font color="Maroon" title="c8">k</font> <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">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:61_1"/><span class="lab"><font color="Green" title="E58">A18</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T26" title="FINSEQ_1:th.26">FINSEQ_1:26</a></span>;<br/>
<a NAME="E7:61_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A19</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E5:61_1"><span class="lab"><font color="Green" title="E57">A17</font></span></a></span>;<br/>
<a NAME="E8:61_1"/><span class="lab"><font color="Green" title="E60">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="txt" href="#E6:61_1"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E9:61_1"/><span class="lab"><font color="Green" title="E61">A21</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E4:61_1"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E10:61_1"/>
<font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E4:61_1"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:61_1"/>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E4:61_1"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="txt" href="#E7:61_1"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="txt" href="#E8:61_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="txt" href="#E9:61_1"><span class="lab"><font color="Green" title="E61">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:61"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <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="c7">F2</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c6">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <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="c7">F2</font> implies <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c6">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_2"/><span class="lab"><font color="Green" title="E53">A22</font></span>: 
<font color="Maroon" title="c8">k</font> <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="c7">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c6">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/>

<a NAME="E2:61_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">F2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A23</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:61"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:61_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A24</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="c6">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></span>;<br/>
<a NAME="E5:61_2"/><span class="lab"><font color="Green" title="E56">A25</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c6">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</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="c6">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E7:61"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E3:61_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></span>;<br/>
<a NAME="E6:61_2"/><span class="lab"><font color="Green" title="E57">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</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="c6">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c6">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_2"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E7:61_2"/><span class="lab"><font color="Green" title="E58">A27</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c6">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E3:61_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E8:61_2"/>
<font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E3:61_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:61_2"/>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.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="c6">F1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E1:61_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E3:61_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E5:61_2"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="txt" href="#E6:61_2"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="txt" href="#E7:61_2"><span class="lab"><font color="Green" title="E58">A27</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:61"/><span class="kw">then </span>
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c7">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:61"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E16:61"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:61"/>
 <a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">C1</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <font color="Maroon" title="c4">C2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <font color="Maroon" title="c3">C1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E6:61"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E10:61"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E13:61"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="funct_7.html#T48" title="FUNCT_7:th.48">FUNCT_7:48</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
