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<span class="kw">let </span><font color="Maroon" title="c1">E</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">u</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <br/> for <font color="Olive" title="b2">TS</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <span class="p1">(<span class="default"><a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="flang_1.html#K4" title="FLANG_1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span><br/> for <font color="Olive" title="b3">R</font> being   <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Olive" title="b2">TS</font>  st <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">R</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">u</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</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">TS</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <span class="p1">(<span class="default"><a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="flang_1.html#K4" title="FLANG_1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span><br/> for <font color="Olive" title="b2">R</font> being   <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Olive" title="b1">TS</font>  st <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">R</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">u</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">TS</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <span class="p1">(<span class="default"><a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="flang_1.html#K4" title="FLANG_1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span>; <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">R</font> being   <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c3">TS</font>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">R</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">u</font></span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">R</font> being   <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c3">TS</font><br/> for <font color="Olive" title="b2">u</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">R</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b2">u</font>;<br/>
<div><span class="lab"><font color="Green" title="E26">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:45_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">R</font> be    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c3">TS</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">u</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive">b<sub>3</sub></font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">u</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45_1_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E2:45_1_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font>
 <span class="kw">and </span><br/><a NAME="E3:45_1_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive">b<sub>2</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_1_1_1"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">u</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_1_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:45_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">u</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c8">u1</font> being    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E3:45_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="flang_1.html#K3" title="FLANG_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c7">e</font></span><a href="flang_1.html#K3" title="FLANG_1:func.3">%&gt;</a></span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T7" target="_self" title="FSM_3:th.7">Th7</a></span>;<br/><a NAME="E4:45_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E3:45_1_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E1:45_1_1_1_2"><span class="lab"><font color="Green" title="E31">A6</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">R9</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c3">TS</font><span class="kw"> such that </span><br/><a NAME="E6:45_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c9">R9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R</font>
 <span class="kw">and </span><br/><a NAME="E7:45_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">R9</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a>, <a class="ref" href="rewrite3.html#T5" title="REWRITE3:th.5">REWRITE3:5</a></span>;<br/><a NAME="E8:45_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">R9</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E7:45_1_1_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E9:45_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">R9</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><a NAME="E10:45_1_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">R9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">R9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E11:45_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">R9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">R9</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E6:45_1_1_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E7:45_1_1_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="rewrite3.html#T1" title="REWRITE3:th.1">REWRITE3:1</a></span>;<br/><a NAME="E12:45_1_1_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">R9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_1_1_1_2"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E13:45_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c9">R9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">R9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite3.html#T1" title="REWRITE3:th.1">REWRITE3:1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_1_1_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="flang_1.html#K3" title="FLANG_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c7">e</font></span><a href="flang_1.html#K3" title="FLANG_1:func.3">%&gt;</a></span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">u1</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E3:45_1_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E3:45_1_1_1_2"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="fsm_3.html#T23" target="_self" title="FSM_3:th.23">Th23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="flang_1.html#K3" title="FLANG_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c7">e</font></span><a href="flang_1.html#K3" title="FLANG_1:func.3">%&gt;</a></span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c8">u1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:45_1_1_1_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E3:45_1_1_1_2"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E6:45_1_1_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E7:45_1_1_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E8:45_1_1_1_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E13:45_1_1_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E11:45_1_1_1_2"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:45_1_1_1_2_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">R9</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E6:45_1_1_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E7:45_1_1_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E13:45_1_1_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E11:45_1_1_1_2"><span class="lab"><font color="Green" title="E37">A12</font></span></a></span>;<br/><a NAME="E3:45_1_1_1_2_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">u1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_1_1_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E4:45_1_1_1_2_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="flang_1.html#K3" title="FLANG_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c7">e</font></span><a href="flang_1.html#K3" title="FLANG_1:func.3">%&gt;</a></span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">u1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D5" title="AFINSQ_1:def.5">AFINSQ_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:45_1_1_1_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_1_2"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:45_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:45"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E3:45"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E26">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:45"/>
for <font color="Olive" title="b1">R</font> being   <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c3">TS</font>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">R</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">R</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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