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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> ;<br/><b>let </b><font color="Maroon" title="c2">N</font> be    <a href="pnproc_1.html#M3" title="PNPROC_1:mode.3">Petri_net</a> of <font color="Maroon" title="c1">P</font>;<br/><b>let </b><font color="Maroon" title="c3">R1</font>, <font color="Maroon" title="c4">R2</font>, <font color="Maroon" title="c5">P1</font>, <font color="Maroon" title="c6">P2</font> be   <a href="pnproc_1.html#NM6" title="PNPROC_1:NM.6">process</a> of <font color="Maroon" title="c2">N</font>;<br/><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>













<b>assume </b><a NAME="E1:120"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K13" title="PNPROC_1:func.13">concur</a> <font color="Maroon" title="c4">R2</font></span>)</span> <a href="pnproc_1.html#K12" title="PNPROC_1:func.12">before</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="pnproc_1.html#K13" title="PNPROC_1:func.13">concur</a> <font color="Maroon" title="c6">P2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">C1</font>, <font color="Maroon" title="c9">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><b> such that </b><br/><a NAME="E3:120"/><i><font color="Green" title="E82">A1</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c9">C2</font> &amp; <font color="Maroon" title="c8">C1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K13" title="PNPROC_1:func.13">concur</a> <font color="Maroon" title="c4">R2</font> &amp; <font color="Maroon" title="c9">C2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font> <a href="pnproc_1.html#K13" title="PNPROC_1:func.13">concur</a> <font color="Maroon" title="c6">P2</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c10">C3</font> being   <a href="pnproc_1.html#NM5" title="PNPROC_1:NM.5">firing-sequence</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E5:120"/><i><font color="Green" title="E83">A2</font></i>: 
( <font color="Maroon" title="c8">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C3</font> &amp;  ex <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">q2</font> being  <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> st <br/>( <font color="Maroon" title="c10">C3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">q2</font> &amp; <font color="Olive" title="b1">q1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">q2</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Olive" title="b1">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Olive" title="b2">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> ) )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E3:120"><i><font color="Green" title="E82">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">q1</font>, <font color="Maroon" title="c12">q2</font> being   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a><b> such that </b><br/><a NAME="E7:120"/><i><font color="Green" title="E84">A3</font></i>: 
( <font color="Maroon" title="c10">C3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">q2</font> &amp; <font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c12">q2</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c12">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E5:120"><i><font color="Green" title="E83">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">C4</font> being   <a href="pnproc_1.html#NM5" title="PNPROC_1:NM.5">firing-sequence</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E9:120"/><i><font color="Green" title="E85">A4</font></i>: 
( <font color="Maroon" title="c9">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">C4</font> &amp;  ex <font color="Olive" title="b1">q3</font>, <font color="Olive" title="b2">q4</font> being  <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> st <br/>( <font color="Maroon" title="c13">C4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q3</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">q4</font> &amp; <font color="Olive" title="b1">q3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">q4</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Olive" title="b1">q3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Olive" title="b2">q4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</font> ) )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E3:120"><i><font color="Green" title="E82">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">q3</font>, <font color="Maroon" title="c15">q4</font> being   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a><b> such that </b><br/><a NAME="E11:120"/><i><font color="Green" title="E86">A5</font></i>: 
( <font color="Maroon" title="c13">C4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q3</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">q4</font> &amp; <font color="Maroon" title="c14">q3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c15">q4</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c14">q3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c15">q4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P2</font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E9:120"><i><font color="Green" title="E85">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">C</font> = <font color="Maroon" title="c8">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c9">C2</font> as   <a href="pnproc_1.html#NM5" title="PNPROC_1:NM.5">firing-sequence</a> of <font color="Maroon" title="c2">N</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">q5</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font>, <font color="Maroon" title="c18">q6</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> ;<br/>
<a NAME="E14:120"/><i><font color="Green" title="E87">A6</font></i>: 
( <font color="Maroon" title="c11">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C1</font> &amp; <font color="Maroon" title="c12">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C1</font> &amp; <font color="Maroon" title="c14">q3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C2</font> &amp; <font color="Maroon" title="c15">q4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C2</font> )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E5:120"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="pnproc_1.html#E7:120"><i><font color="Green" title="E84">A3</font></i></a>, <a class="txt" href="pnproc_1.html#E9:120"><i><font color="Green" title="E85">A4</font></i></a>, <a class="txt" href="pnproc_1.html#E11:120"><i><font color="Green" title="E86">A5</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E15:120"/><b>then </b><i><font color="Green" title="E88">A7</font></i>: 
( <font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">q5</font> &amp; <font color="Maroon" title="c12">q2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c18">q6</font> )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E93"><i><font color="Green" title="E66">Lm9</font></i></a></i>;<br/>
<a NAME="E16:120"/>
<font color="Maroon" title="c8">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c9">C2</font>
 
<b>by </b><i><a class="ref" href="finseq_6.html#T12" title="FINSEQ_6:th.12">FINSEQ_6:12</a></i>;<br/>
<a NAME="E17:120"/><b>then </b><i><font color="Green" title="E89">A8</font></i>: 
( <font color="Maroon" title="c11">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c9">C2</font> &amp; <font color="Maroon" title="c12">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c9">C2</font> )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E14:120"><i><font color="Green" title="E87">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">ss</font> = <font color="Maroon" title="c9">C2</font> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> ;<br/>
<a NAME="E19:120"/><i><font color="Green" title="E90">A9</font></i>: 
( <font color="Maroon" title="c17">q5</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c19">ss</font> &amp; <font color="Maroon" title="c18">q6</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c19">ss</font> )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E14:120"><i><font color="Green" title="E87">A6</font></i></a>, <a class="ref" href="pnproc_1.html#T70" target="_self" title="PNPROC_1:th.70">Th70</a></i>;<br/>
<a NAME="E20:120"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c9">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c9">C2</font>
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T71" target="_self" title="PNPROC_1:th.71">Th71</a></i>;<br/>
<a NAME="E21:120"/><b>then </b>
( <font color="Maroon" title="c17">q5</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c9">C2</font> &amp; <font color="Maroon" title="c18">q6</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c9">C2</font> )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E19:120"><i><font color="Green" title="E90">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E22:120"/><b>then </b>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">q5</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">q2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">q6</font> )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E15:120"><i><font color="Green" title="E88">A7</font></i></a>, <a class="txt" href="pnproc_1.html#E17:120"><i><font color="Green" title="E89">A8</font></i></a>, <a class="ref" href="pnproc_1.html#T10" target="_self" title="PNPROC_1:th.10">Th10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">ss1</font> = <font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c17">q5</font>, <font color="Maroon" title="c21">ss2</font> = <font color="Maroon" title="c12">q2</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c18">q6</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="ref" href="grfunc_1.html#T31" title="GRFUNC_1:th.31">GRFUNC_1:31</a></i>;<br/>
<a NAME="E24:120"/><i><font color="Green" title="E91">A10</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">C</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="c16">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E25:120"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c20">ss1</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> <font color="Maroon" title="c16">C</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c21">ss2</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> <font color="Maroon" title="c16">C</font> )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E14:120"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="pnproc_1.html#E92"><i><font color="Green" title="E65">Lm8</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">ss1</font> = <font color="Maroon" title="c20">ss1</font>, <font color="Maroon" title="c23">ss2</font> = <font color="Maroon" title="c21">ss2</font> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> <b>by </b><i><a class="txt" href="pnproc_1.html#E24:120"><i><font color="Green" title="E91">A10</font></i></a>, <a class="ref" href="finseq_1.html#D12" title="FINSEQ_1:def.12">FINSEQ_1:def 12</a></i>;<br/>
<i><font color="Green" title="E92">A11</font></i>: <font color="Maroon" title="c22">ss1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c23">ss2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">q2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c12">q2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon" title="c8">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pnproc_1.html#E5:120"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="pnproc_1.html#E7:120"><i><font color="Green" title="E84">A3</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon" title="c8">C1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c9">C2</font></span>)</span>
<b>by </b><i><a class="txt" href="pnproc_1.html#E9:120"><i><font color="Green" title="E85">A4</font></i></a>, <a class="txt" href="pnproc_1.html#E11:120"><i><font color="Green" title="E86">A5</font></i></a>, <a class="ref" href="pnproc_1.html#T73" target="_self" title="PNPROC_1:th.73">Th73</a></i>
<br/>.= 
<font color="Maroon" title="c16">C</font>
<b>by </b><i><a class="ref" href="pnproc_1.html#T64" target="_self" title="PNPROC_1:th.64">Th64</a></i>
;<br/>
<a NAME="E28:120"/><i><font color="Green" title="E93">A12</font></i>: 
<font color="Maroon" title="c22">ss1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c23">ss2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_2"><b>proof </b></a><div class="add">

<a NAME="E1:120_2"/>
( <font color="Maroon" title="c22">ss1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">q2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">q2</font></span>)</span> &amp; <font color="Maroon" title="c22">ss1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E2:120_2"/><b>then </b><i><font color="Green" title="E94">A13</font></i>: 
<font color="Maroon" title="c22">ss1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c23">ss2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">q2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">q2</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E3:120_2"/><i><font color="Green" title="E95">A14</font></i>: 
<font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E7:120"><i><font color="Green" title="E84">A3</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E4:120_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c12">q2</font> &amp; <font color="Maroon" title="c11">q1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</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">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font> )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E14:120"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="pnproc_1.html#E93"><i><font color="Green" title="E66">Lm9</font></i></a></i>;<br/>
<a NAME="E5:120_2"/><b>then </b><i><font color="Green" title="E96">A15</font></i>: 
( <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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">q2</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="c11">q1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E6:120_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">q3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">q4</font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E11:120"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="pnproc_1.html#E14:120"><i><font color="Green" title="E87">A6</font></i></a>, <a class="ref" href="pnproc_1.html#T10" target="_self" title="PNPROC_1:th.10">Th10</a></i>;<br/>
<a NAME="E7:120_2"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</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="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T72" target="_self" title="PNPROC_1:th.72">Th72</a></i>;<br/>
<a NAME="E8:120_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c14">q3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</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">C1</font></span>)</span> <a href="pnproc_1.html#K14" title="PNPROC_1:func.14">Shift</a> <font color="Maroon" title="c15">q4</font>
 
<b>by </b><i><a class="ref" href="pnproc_1.html#T8" target="_self" title="PNPROC_1:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E9:120_2"/>
<font color="Maroon" title="c22">ss1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c23">ss2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="pnproc_1.html#E2:120_2"><i><font color="Green" title="E94">A13</font></i></a>, <a class="txt" href="pnproc_1.html#E3:120_2"><i><font color="Green" title="E95">A14</font></i></a>, <a class="txt" href="pnproc_1.html#E5:120_2"><i><font color="Green" title="E96">A15</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c24">ss3</font> being   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a><b> such that </b><br/><a NAME="E30:120"/><i><font color="Green" title="E94">A16</font></i>: 
( <font color="Maroon" title="c24">ss3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">ss1</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c11">q1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c14">q3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c24">ss3</font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E14:120"><i><font color="Green" title="E87">A6</font></i></a>, <a class="ref" href="pnproc_1.html#T82" target="_self" title="PNPROC_1:th.82">Th82</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">ss4</font> being   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a><b> such that </b><br/><a NAME="E32:120"/><i><font color="Green" title="E95">A17</font></i>: 
( <font color="Maroon" title="c25">ss4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">ss2</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c12">q2</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c15">q4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c25">ss4</font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E14:120"><i><font color="Green" title="E87">A6</font></i></a>, <a class="ref" href="pnproc_1.html#T82" target="_self" title="PNPROC_1:th.82">Th82</a></i>;<br/>
<a NAME="E33:120"/><i><font color="Green" title="E96">A18</font></i>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c22">ss1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K12" title="PNPROC_1:func.12">before</a> <font color="Maroon" title="c5">P1</font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E7:120"><i><font color="Green" title="E84">A3</font></i></a>, <a class="txt" href="pnproc_1.html#E11:120"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="pnproc_1.html#E30:120"><i><font color="Green" title="E94">A16</font></i></a></i>;<br/>
<a NAME="E34:120"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c23">ss2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K12" title="PNPROC_1:func.12">before</a> <font color="Maroon" title="c6">P2</font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E7:120"><i><font color="Green" title="E84">A3</font></i></a>, <a class="txt" href="pnproc_1.html#E11:120"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="pnproc_1.html#E32:120"><i><font color="Green" title="E95">A17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E35:120"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R1</font> <a href="pnproc_1.html#K12" title="PNPROC_1:func.12">before</a> <font color="Maroon" title="c5">P1</font></span>)</span> <a href="pnproc_1.html#K13" title="PNPROC_1:func.13">concur</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R2</font> <a href="pnproc_1.html#K12" title="PNPROC_1:func.12">before</a> <font color="Maroon" title="c6">P2</font></span>)</span>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E3:120"><i><font color="Green" title="E82">A1</font></i></a>, <a class="txt" href="pnproc_1.html#E27:120"><i><font color="Green" title="E92">A11</font></i></a>, <a class="txt" href="pnproc_1.html#E28:120"><i><font color="Green" title="E93">A12</font></i></a>, <a class="txt" href="pnproc_1.html#E33:120"><i><font color="Green" title="E96">A18</font></i></a></i>;<br/>


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