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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N</font> being   <a href="pnproc_1.html#M4" title="PNPROC_1:mode.4">Petri_net</a> of <font color="Maroon" title="c1">P</font><br/> for <font color="Olive" title="b2">e1</font>, <font color="Olive" title="b3">e2</font> being   <a href="pnproc_1.html#M5" title="PNPROC_1:mode.5">Element</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"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Olive" title="b2">e1</font>,<font color="Olive" title="b3">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Olive" title="b3">e2</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#K7" title="PNPROC_1:func.7">fire</a> <font color="Olive" title="b2">e1</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">N</font> be    <a href="pnproc_1.html#M4" title="PNPROC_1:mode.4">Petri_net</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being   <a href="pnproc_1.html#M5" title="PNPROC_1:mode.5">Element</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"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Olive" title="b2">e2</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#K7" title="PNPROC_1:func.7">fire</a> <font color="Olive" title="b1">e1</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">e1</font>, <font color="Maroon" title="c4">e2</font> be    <a href="pnproc_1.html#M5" title="PNPROC_1:mode.5">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c4">e2</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#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c3">e1</font></span>)</span></span><br/>







<b>consider </b><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><b> such that </b><br/><a NAME="E2:64"/><i><font color="Green" title="E43">A1</font></i>: 
 <a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></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#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">P</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>
 <b>and </b><br/><a NAME="E3:64"/><i><font color="Green" title="E44">A2</font></i>: 
 <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"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:64"/><i><font color="Green" title="E45">A3</font></i>: 
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"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></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"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#D11" target="_self" title="PNPROC_1:def.11">Def11</a></i>;<br/>
<a NAME="E5:64"/><i><font color="Green" title="E46">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e1</font> &amp; <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E6:64"/><b>then </b><i><font color="Green" title="E47">A5</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,2</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:64"/><i><font color="Green" title="E48">A6</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,2</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,2</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E8:64"/><b>then </b><i><font color="Green" title="E49">A7</font></i>: 
( <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )
 
<b>by </b><i><a class="txt" href="#E6:64"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E9:64"/><b>then </b><i><font color="Green" title="E50">A8</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c3">e1</font>
 
<b>by </b><i><a class="txt" href="#E4:64"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="#E5:64"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="#E6:64"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="#E7:64"><i><font color="Green" title="E48">A6</font></i></a></i>;<br/>
<a NAME="E10:64"/>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c4">e2</font>
 
<b>by </b><i><a class="txt" href="#E4:64"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="#E5:64"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="#E6:64"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="#E7:64"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="#E8:64"><i><font color="Green" title="E49">A7</font></i></a></i>;<br/>
<a NAME="E11:64"/><b>then </b><i><font color="Green" title="E51">A9</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c3">e1</font></span>)</span>,<span class="p2">(<span class="default"><a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c4">e2</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E3:64"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="#E5:64"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="#E9:64"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E12:64"/>
<span class="p1">(<span class="default"><a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c3">e1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">P</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></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> <font color="Maroon" title="c3">e1</font>
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T38" target="_self" title="PNPROC_1:th.38">Th38</a></i>;<br/>
<a NAME="E13:64"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c4">e2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c3">e1</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">P</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c4">e2</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#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c3">e1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/>
<b>hence </b><a NAME="E14:64"/>
 <a href="pnproc_1.html#K8" title="PNPROC_1:func.8">fire</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c4">e2</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#K7" title="PNPROC_1:func.7">fire</a> <font color="Maroon" title="c3">e1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:64"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="#E11:64"><i><font color="Green" title="E51">A9</font></i></a>, <a class="ref" href="funct_7.html#T53" title="FUNCT_7:th.53">FUNCT_7:53</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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