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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>, <font color="Olive" title="b2">v</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="b3">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="b4">p</font>, <font color="Olive" title="b5">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b3">TS</font>  st <font color="Olive" title="b4">p</font>,<font color="Olive" title="b1">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b2">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b5">q</font>,<font color="Olive" title="b3">TS</font> holds <br/> ex <font color="Olive" title="b6">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b3">TS</font> st <br/>( <font color="Olive" title="b4">p</font>,<font color="Olive" title="b1">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b6">r</font>,<font color="Olive" title="b3">TS</font> &amp; <font color="Olive" title="b6">r</font>,<font color="Olive" title="b2">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b5">q</font>,<font color="Olive" title="b3">TS</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</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">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">TS</font>  st <font color="Olive" title="b2">p</font>,<font color="Maroon" title="c2">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b3">q</font>,<font color="Olive" title="b1">TS</font> holds <br/> ex <font color="Olive" title="b4">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">TS</font> st <br/>( <font color="Olive" title="b2">p</font>,<font color="Maroon" title="c2">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b4">r</font>,<font color="Olive" title="b1">TS</font> &amp; <font color="Olive" title="b4">r</font>,<font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b3">q</font>,<font color="Olive" title="b1">TS</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">TS</font>  st <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c4">TS</font> holds <br/> ex <font color="Olive" title="b3">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">TS</font> st <br/>( <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b3">r</font>,<font color="Maroon" title="c4">TS</font> &amp; <font color="Olive" title="b3">r</font>,<font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c4">TS</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">TS</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" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">TS</font> implies  ex <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">TS</font> st <br/>( <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c4">TS</font> &amp; <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">TS</font> ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">TS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">TS</font> st <br/>( <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c4">TS</font> &amp; <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">TS</font> )</span><br/>

<a NAME="E2:51"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R3" title="REWRITE3:pred.3">==&gt;*</a> <font color="Maroon" title="c6">q</font>, <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">TS</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite3.html#D7" title="REWRITE3:def.7">REWRITE3:def 7</a></span>;<br/>
<a NAME="E3:51"/><span class="kw">then </span>
 <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c4">TS</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">v</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">q</font>,<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="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite3.html#D6" title="REWRITE3:def.6">REWRITE3:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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="c4">TS</font><span class="kw"> such that </span><br/><a NAME="E5:51"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c7">R</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="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">v</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:51"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c7">R</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="c7">R</font></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="Maroon" title="c6">q</font>,<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="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></span>;<br/>
<a NAME="E7:51"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c7">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E8:51"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E33">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">then </span><span class="kw">consider </span><font color="Maroon" title="c8">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:51"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon" title="c8">l</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="c7">R</font>
 <span class="kw">and </span><br/><a NAME="E11:51"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">l</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="c3">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="fsm_3.html#T25" target="_self" title="FSM_3:th.25">Th25</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_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="E1:51_1"/>
( <font color="Maroon" title="c8">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c7">R</font> or  not <font color="Maroon" title="c8">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c7">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><span class="kw">suppose </span></a><a NAME="E1:51_1_1"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<font color="Maroon" title="c8">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c7">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">TS</font> st <br/>( <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c4">TS</font> &amp; <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">TS</font> )</span><br/><div class="add"><a NAME="E2:51_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="rewrite3.html#T49" title="REWRITE3:th.49">REWRITE3:49</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">l</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">TS</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/><a NAME="E4:51_1_1"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<font color="Maroon" title="c7">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">l</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="Maroon" title="c9">r</font>,<font color="Maroon" title="c3">v</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="#E10:51"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E1:51_1_1"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="rewrite3.html#T48" title="REWRITE3:th.48">REWRITE3:48</a></span>;<br/><a NAME="E5:51_1_1"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<font color="Maroon" title="c8">l</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="c7">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c9">r</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" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c4">TS</font> &amp; <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">TS</font> )</span><br/><a NAME="E6:51_1_1"/><span class="lab"><font color="Green" title="E41">A10</font></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="c7">R</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">l</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">l</font> = <font color="Maroon" title="c8">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E8:51_1_1"/>
 <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c4">TS</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c7">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E6:51_1_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="rewrite1.html#T18" title="REWRITE1:th.18">REWRITE1:18</a></span>;<br/><a NAME="E9:51_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R3" title="REWRITE3:pred.3">==&gt;*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E4:51_1_1"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="rewrite3.html#D6" title="REWRITE3:def.6">REWRITE3:def 6</a></span>;<br/><a NAME="E10:51_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R3" title="REWRITE3:pred.3">==&gt;*</a> <font color="Maroon" title="c9">r</font>,<span class="p1">(<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> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></span>;<br/><a NAME="E11:51_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="rewrite3.html#R3" title="REWRITE3:pred.3">==&gt;*</a> <font color="Maroon" title="c9">r</font>, <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T28" target="_self" title="FSM_3:th.28">Th28</a></span>;<br/><span class="kw">hence </span><a NAME="E12:51_1_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c4">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite3.html#D7" title="REWRITE3:def.7">REWRITE3:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">TS</font></span><br/><a NAME="E13:51_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c7">R</font>
 <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="E14:51_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">R</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="c7">R</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="E15:51_1_1"/><span class="kw">then </span>
 <a href="rewrite3.html#K1" title="REWRITE3:func.1">==&gt;.-relation</a> <font color="Maroon" title="c4">TS</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c7">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">l</font>,<font color="Maroon" title="c7">R</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="c7">R</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E5:51_1_1"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="rewrite1.html#T18" title="REWRITE1:th.18">REWRITE1:18</a></span>;<br/><a NAME="E16:51_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R3" title="REWRITE3:pred.3">==&gt;*</a> <font color="Maroon" title="c6">q</font>, <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c4">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E4:51_1_1"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="rewrite3.html#D6" title="REWRITE3:def.6">REWRITE3:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E17:51_1_1"/>
<font color="Maroon" title="c9">r</font>,<font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite3.html#D7" title="REWRITE3:def.7">REWRITE3:def 7</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="51_1_2"><span class="kw">suppose </span></a><a NAME="E1:51_1_2"/>
not <font color="Maroon" title="c8">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c7">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">TS</font> st <br/>( <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c4">TS</font> &amp; <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">TS</font> )</span><br/><div class="add"><a NAME="E2:51_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c3">v</font> <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="#E7:51"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="rewrite3.html#T3" title="REWRITE3:th.3">REWRITE3:3</a></span>;<br/><a NAME="E3:51_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></span>;<br/><span class="kw">hence </span><a NAME="E4:51_1_2"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">TS</font> st <br/>( <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c4">TS</font> &amp; <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c3">v</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">TS</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1_2"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="ref" href="rewrite3.html#T95" title="REWRITE3:th.95">REWRITE3:95</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>
