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

<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="Red">::  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">e1</font>, <font color="Olive" title="b3">e2</font> being   <a href="pnproc_1.html#M3" title="PNPROC_1:mode.3">Element</a> of <font color="Olive" title="b1">N</font> holds  <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Olive" title="b3">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b2">e1</font>,<font color="Olive" title="b3">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>,<span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b3">e2</font>,<font color="Olive" title="b2">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></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="Red">::  thesis: </font></span></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#M3" title="PNPROC_1:mode.3">Element</a> of <font color="Maroon" title="c2">N</font> holds  <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>,<span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Olive" title="b2">e2</font>,<font color="Olive" title="b1">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">e1</font>, <font color="Maroon" title="c4">e2</font> be    <a href="pnproc_1.html#M3" title="PNPROC_1:mode.3">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>,<span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font>,<font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">C1</font> = <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">C2</font> = <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">R1</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">R2</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">thus </span><a NAME="E1:79"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>,<span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font>,<font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>,<span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font>,<font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c9">x</font> <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"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>,<span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font>,<font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <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"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>,<span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font>,<font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">C</font> being   <a href="pnproc_1.html#NM5" title="PNPROC_1:NM.5">firing-sequence</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E3:79_1"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font>
 <span class="kw">and </span><br/><a NAME="E4:79_1"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
 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">C</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> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &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> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 ;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:79_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c10">C</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>
 <span class="kw">and </span><br/><a NAME="E7:79_1"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:79_1"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
 <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> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E9:79_1"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
 <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> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>;<br/>
<a NAME="E10:79_1"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:79_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E11:79_1"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c12">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:79_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">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> <span class="kw"> such that </span><br/><a NAME="E13:79_1"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
<font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:79_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="pnproc_1.html#T4" target="_self" title="PNPROC_1:th.4">Th4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">j</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> <span class="kw"> such that </span><br/><a NAME="E15:79_1"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
<font color="Maroon" title="c12">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">j</font>,<font color="Maroon" title="c4">e2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:79_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="pnproc_1.html#T4" target="_self" title="PNPROC_1:th.4">Th4</a></span>;<br/>
<a NAME="E16:79_1"/><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:79_1"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E17:79_1"/><span class="lab"><font color="Green" title="E57">A12</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">j</font>,<font color="Maroon" title="c4">e2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:79_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E18:79_1"/><span class="lab"><font color="Green" title="E58">A13</font></span>: 
<font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">j</font>,<font color="Maroon" title="c4">e2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E13:79_1"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E15:79_1"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></span>;<br/>
<a NAME="E19:79_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e2</font> ) or ( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T5" target="_self" title="PNPROC_1:th.5">Th5</a></span>;<br/>
<a NAME="E20:79_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> or <font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font>,<font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:79_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E16:79_1"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="txt" href="#E17:79_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E18:79_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="ref" href="pnproc_1.html#T6" target="_self" title="PNPROC_1:th.6">Th6</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:79_1"/>
<font color="Maroon" title="c9">x</font> <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"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>,<span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font>,<font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <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"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>,<span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font>,<font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:79"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<font color="Maroon" title="c9">x</font> <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"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>,<span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font>,<font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:79_2"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font>,<font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1"><span class="kw">suppose </span></a><a NAME="E1:79_2_1"/><span class="lab"><font color="Green" title="E47">A15</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font>,<font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A16</font></span>: <font color="Maroon" title="c9">x</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<font color="Maroon" title="c4">e2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T6" target="_self" title="PNPROC_1:th.6">Th6</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<font color="Maroon" title="c4">e2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">q1</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>, <font color="Maroon" title="c11">q2</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<font color="Maroon" title="c4">e2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T1" target="_self" title="PNPROC_1:th.1">Th1</a></span>;<br/><a NAME="E4:79_2_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<font color="Maroon" title="c4">e2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E5:79_2_1"/><span class="kw">then </span>
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E6:79_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A17</font></span>: 
<font color="Maroon" title="c10">q1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/><a NAME="E7:79_2_1"/><span class="lab"><font color="Green" title="E50">A18</font></span>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c10">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T3" target="_self" title="PNPROC_1:th.3">Th3</a></span>;<br/><a NAME="E8:79_2_1"/><span class="lab"><font color="Green" title="E51">A19</font></span>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c11">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T3" target="_self" title="PNPROC_1:th.3">Th3</a></span>;<br/><a NAME="E9:79_2_1"/><span class="lab"><font color="Green" title="E52">A20</font></span>: 
<span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E10:79_2_1"/>
<span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E11:79_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_1"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E2:79_2_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E6:79_2_1"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="txt" href="#E7:79_2_1"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="txt" href="#E8:79_2_1"><span class="lab"><font color="Green" title="E51">A19</font></span></a>, <a class="txt" href="#E9:79_2_1"><span class="lab"><font color="Green" title="E52">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_2"><span class="kw">suppose </span></a><a NAME="E1:79_2_2"/><span class="lab"><font color="Green" title="E47">A21</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font>,<font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A22</font></span>: <font color="Maroon" title="c9">x</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<font color="Maroon" title="c4">e2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T6" target="_self" title="PNPROC_1:th.6">Th6</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<font color="Maroon" title="c4">e2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">q1</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>, <font color="Maroon" title="c11">q2</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<font color="Maroon" title="c4">e2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T1" target="_self" title="PNPROC_1:th.1">Th1</a></span>;<br/><a NAME="E4:79_2_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<font color="Maroon" title="c4">e2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E5:79_2_2"/><span class="kw">then </span>
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<font color="Maroon" title="c3">e1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E6:79_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A23</font></span>: 
<font color="Maroon" title="c10">q1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></span>;<br/><a NAME="E7:79_2_2"/><span class="lab"><font color="Green" title="E50">A24</font></span>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c10">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T3" target="_self" title="PNPROC_1:th.3">Th3</a></span>;<br/><a NAME="E8:79_2_2"/><span class="lab"><font color="Green" title="E51">A25</font></span>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c11">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T3" target="_self" title="PNPROC_1:th.3">Th3</a></span>;<br/><a NAME="E9:79_2_2"/><span class="lab"><font color="Green" title="E52">A26</font></span>: 
<span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E10:79_2_2"/>
<span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E11:79_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">e1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="pnproc_1.html#K11" title="PNPROC_1:func.11">concur</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">e2</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2_2"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E2:79_2_2"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E6:79_2_2"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="txt" href="#E7:79_2_2"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E8:79_2_2"><span class="lab"><font color="Green" title="E51">A25</font></span></a>, <a class="txt" href="#E9:79_2_2"><span class="lab"><font color="Green" title="E52">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
