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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being   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> <br/>  for <font color="Olive" title="b2">u</font>, <font color="Olive" title="b3">w</font>, <font color="Olive" title="b4">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <br/>  for <font color="Olive" title="b5">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="Olive" title="b1">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="Olive" title="b1">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span><br/>  for <font color="Olive" title="b6">p</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="b5">TS</font>  st <font color="Olive" title="b6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b3">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Olive" title="b4">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b3">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/> ex <font color="Olive" title="b7">q</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="b5">TS</font> st <br/>( <font color="Olive" title="b7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Olive" title="b2">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b7">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b7">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Olive" title="b4">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">w</font>, <font color="Olive" title="b3">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <br/>  for <font color="Olive" title="b4">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="c3">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="c3">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span><br/>  for <font color="Olive" title="b5">p</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="b4">TS</font>  st <font color="Olive" title="b5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b2">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b2">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/> ex <font color="Olive" title="b6">q</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="b4">TS</font> st <br/>( <font color="Olive" title="b6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Olive" title="b3">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">w</font>, <font color="Maroon" title="c6">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">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="firebrick">::  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="c3">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="c3">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">p</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 <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/> ex <font color="Olive" title="b3">q</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 <br/>( <font color="Olive" title="b3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">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="c3">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="c3">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</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="c7">TS</font>  st <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/> ex <font color="Olive" title="b2">q</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="c7">TS</font> st <br/>( <font color="Olive" title="b2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">p</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="c7">TS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies  ex <font color="Olive" title="b1">q</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="c7">TS</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q</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="c7">TS</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<a NAME="E3:47"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</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#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:47"/><span class="kw">then </span>
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="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E32">A4</font></span>:  <a href="rewrite3.html#K2" title="REWRITE3:func.2">dim2</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c3">E</font>) = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="rewrite3.html#T51" title="REWRITE3:th.51">REWRITE3:51</a></span>
<br/>.= 
<font color="Maroon" title="c4">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><a href="fsm_3.html#K7" title="FSM_3:func.7">chop</a> (<span class="p3">(<span class="default"><a href="rewrite3.html#K2" title="REWRITE3:func.2">dim2</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<font color="Maroon" title="c3">E</font>)</span>)</span>,<font color="Maroon" title="c5">w</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E7:47"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font>
 <span class="kw">and </span><br/><a NAME="E8:47"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">q</font> holds <br/><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></span>
<a NAME="E9:47"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">q</font> &amp; <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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">q</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c7">TS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">q</font> &amp; <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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">q</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c7">TS</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47_2"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">q</font>
 <span class="kw">and </span><br/><a NAME="E2:47_2"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c7">TS</font></span><br/>

<a NAME="E3:47_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:47_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">v1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E6:47_2"/><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></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="c11">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="rewrite3.html#T52" title="REWRITE3:th.52">REWRITE3:52</a></span>;<br/>
<a NAME="E7:47_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E8:47_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A12</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">v2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E10:47_2"/><span class="lab"><font color="Green" title="E41">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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> <font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="rewrite3.html#T52" title="REWRITE3:th.52">REWRITE3:52</a></span>;<br/>
<a NAME="E11:47_2"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c7">TS</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="txt" href="#E8:47_2"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></span>;<br/>
<a NAME="E12:47_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c7">TS</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="txt" href="#E8:47_2"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E10:47_2"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="rewrite3.html#T48" title="REWRITE3:th.48">REWRITE3:48</a></span>;<br/>
<a NAME="E13:47_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c7">TS</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="txt" href="#E8:47_2"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E6:47_2"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="ref" href="rewrite3.html#T48" title="REWRITE3:th.48">REWRITE3:48</a></span>;<br/>
<a NAME="E14:47_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Maroon" title="c11">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,<font color="Maroon" title="c7">TS</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite3.html#D4" title="REWRITE3:def.4">REWRITE3:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">u1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E16:47_2"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Maroon" title="c13">u1</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Maroon" title="c7">TS</font>
 <span class="kw">and </span><br/><a NAME="E17:47_2"/><span class="lab"><font color="Green" title="E44">A16</font></span>: 
<font color="Maroon" title="c11">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">u1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite3.html#T22" title="REWRITE3:th.22">REWRITE3:22</a></span>;<br/>
<a NAME="E18:47_2"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
 ex <font color="Olive" title="b1">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">TS</font> ex <font color="Olive" title="b2">w1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  ex <font color="Olive" title="b3">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">TS</font> ex <font color="Olive" title="b4">w2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  st <br/>( <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">r1</font>,<font color="Olive" title="b2">w1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <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> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">r2</font>,<font color="Olive" title="b4">w2</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="txt" href="#E11:47_2"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="ref" href="rewrite3.html#T31" title="REWRITE3:th.31">REWRITE3:31</a></span>;<br/>
<a NAME="E19:47_2"/><span class="kw">then </span>
 <a href="rewrite3.html#K2" title="REWRITE3:func.2">dim2</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c3">E</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47_2"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="rewrite3.html#D5" title="REWRITE3:def.5">REWRITE3:def 5</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E46">A18</font></span>: <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><a href="fsm_3.html#K7" title="FSM_3:func.7">chop</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">w</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E2:47_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><a href="fsm_3.html#K7" title="FSM_3:func.7">chop</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,<font color="Maroon" title="c5">w</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T27" title="AFINSQ_1:th.27">AFINSQ_1:27</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#D9" target="_self" title="FSM_3:def.9">Def9</a></span>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font> = 
<font color="Maroon" title="c13">u1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:47_2"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="afinsq_1.html#T27" title="AFINSQ_1:th.27">AFINSQ_1:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">u1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c12">v2</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T27" title="AFINSQ_1:th.27">AFINSQ_1:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">u1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c12">v2</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T27" title="AFINSQ_1:th.27">AFINSQ_1:27</a></span>
;<br/>
<span class="kw">then </span><font color="Maroon" title="c11">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">u1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c12">v2</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T28" title="AFINSQ_1:th.28">AFINSQ_1:28</a></span>
<br/>.= 
<font color="Maroon" title="c13">u1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T27" title="AFINSQ_1:th.27">AFINSQ_1:27</a></span>
;<br/>
<a NAME="E23:47_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Maroon" title="c11">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font> <a href="rewrite3.html#R2" title="REWRITE3:pred.2">==&gt;.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ,<font color="Maroon" title="c12">v2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">TS</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:47_2"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="rewrite3.html#D3" title="REWRITE3:def.3">REWRITE3:def 3</a></span>;<br/>
<a NAME="E24:47_2"/>
 <a href="rewrite3.html#K2" title="REWRITE3:func.2">dim2</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<font color="Maroon" title="c3">E</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47_2"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="txt" href="#E18:47_2"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="rewrite3.html#D5" title="REWRITE3:def.5">REWRITE3:def 5</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><a href="fsm_3.html#K7" title="FSM_3:func.7">chop</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c11">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">w</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E1:47_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><a href="fsm_3.html#K7" title="FSM_3:func.7">chop</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,<font color="Maroon" title="c5">w</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T27" title="AFINSQ_1:th.27">AFINSQ_1:27</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">v</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#D9" target="_self" title="FSM_3:def.9">Def9</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E26:47_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c7">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:47_2"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="txt" href="#E20:47_2"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="ref" href="rewrite3.html#D4" title="REWRITE3:def.4">REWRITE3:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:47"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</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="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E36">A20</font></span>:  <a href="rewrite3.html#K2" title="REWRITE3:func.2">dim2</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</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="c8">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">E</font>) = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</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="c8">p</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="rewrite3.html#T51" title="REWRITE3:th.51">REWRITE3:51</a></span>
<br/>.= 
<font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c9">q</font> as    <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="c7">TS</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E9:47"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></span>;<br/>
<a NAME="E13:47"/>
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="c10">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E37">A21</font></span>: <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><a href="fsm_3.html#K7" title="FSM_3:func.7">chop</a> (<span class="p3">(<span class="default"><a href="rewrite3.html#K2" title="REWRITE3:func.2">dim2</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c3">E</font>)</span>)</span>,<font color="Maroon" title="c5">w</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47"><span class="lab"><font color="Green" title="E34">A6</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="fsm_3.html#K7" title="FSM_3:func.7">chop</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,<font color="Maroon" title="c5">w</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#D9" target="_self" title="FSM_3:def.9">Def9</a></span>
;<br/>
<a NAME="E15:47"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</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="c10">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">q</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><a href="fsm_3.html#K7" title="FSM_3:func.7">chop</a> (<span class="p3">(<span class="default"><a href="rewrite3.html#K2" title="REWRITE3:func.2">dim2</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">E</font>)</span>)</span>,<font color="Maroon" title="c5">w</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E8:47"><span class="lab"><font color="Green" title="E34">A6</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><a href="fsm_3.html#K7" title="FSM_3:func.7">chop</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">w</font></span>)</span>,<font color="Maroon" title="c5">w</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E11:47"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#D9" target="_self" title="FSM_3:def.9">Def9</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E17:47"/>
 ex <font color="Olive" title="b1">q</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="c7">TS</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c4">u</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:47"><span class="lab"><font color="Green" title="E37">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
