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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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">w</font>, <font color="Olive" title="b3">v</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">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="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="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> holds  <font color="Olive" title="b2">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Olive" title="b4">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">w</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="c2">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="c2">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="c2">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span> holds  <font color="Olive" title="b1">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Olive" title="b3">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">w</font>, <font color="Maroon" title="c4">v</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">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="c2">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="c2">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span> holds  <font color="Maroon" title="c3">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Olive" title="b1">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">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="c2">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="c2">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"> <font color="Maroon" title="c3">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>









<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">u</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>   st  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/>for <font color="Olive" title="b2">v</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  holds  <font color="Olive" title="b1">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E1:54"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
not  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <span class="p2">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T8" target="_self" title="FSM_3:th.8">Th8</a></span>;<br/>
<a NAME="E2:54"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">u</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">v</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  holds  <font color="Maroon" title="c6">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  holds  <font color="Maroon" title="c6">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<a NAME="E2:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c6">u</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="c2">E</font>
 
;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">v</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">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"> <font color="Maroon" title="c6">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">vso</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#D1" target="_self" title="FSM_3:def.1">Def1</a></span>;<br/>
<a NAME="E4:54_1"/>
<font color="Maroon" title="c6">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c8">vso</font>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vso</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="rewrite3.html#T104" title="REWRITE3:th.104">REWRITE3:104</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:54_1"/>
<font color="Maroon" title="c6">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</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="lab"><font color="Green" title="E37">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:54_2"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">u</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies for <font color="Olive" title="b1">v</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  holds  <font color="Olive">b<sub>2</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:54_2_1"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  holds  <font color="Olive">b<sub>2</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">v</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">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"> <font color="Olive">b<sub>1</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_2_1_1"/>
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:54_2_1_1_1"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_2_1_1_1_1"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E1:54_2_1_1_1"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:54_2_1_1_1_1_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:54_2_1_1_1_1_1"/>
<font color="Maroon" title="c7">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E36">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:54_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E42">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:54_2_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">P</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#D1" target="_self" title="FSM_3:def.1">Def1</a></span>;<br/><a NAME="E3:54_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:54_2_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span>,<font color="Maroon" title="c5">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T30" target="_self" title="FSM_3:th.30">Th30</a></span>;<br/><a NAME="E5:54_2_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A11</font></span>: 
<font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font>,<font color="Maroon" title="c7">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c9">x</font>, <a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="fsm_3.html#T14" target="_self" title="FSM_3:th.14">Th14</a></span>;<br/><a NAME="E6:54_2_1_1_1_1_2_1"/>
( <font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> &amp; <font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#D1" target="_self" title="FSM_3:def.1">Def1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E7:54_2_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_2_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="rewrite3.html#T103" title="REWRITE3:th.103">REWRITE3:103</a></span>;<br/><span class="kw">hence </span><a NAME="E8:54_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</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="hs2(this)" href="javascript:()" title="54_2_1_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:54_2_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">P</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> ;<br/><span class="kw">consider </span><font color="Maroon" title="c11">Q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:54_2_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
( <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c11">Q</font>,<font color="Maroon" title="c7">u</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c10">P</font>, <a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="rewrite3.html#T103" title="REWRITE3:th.103">REWRITE3:103</a></span>;<br/><a NAME="E5:54_2_1_1_1_1_2_2"/>
( <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font> &amp; <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c11">Q</font>,<font color="Maroon" title="c5">TS</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E4:54_2_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="fsm_3.html#T14" target="_self" title="FSM_3:th.14">Th14</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E6:54_2_1_1_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T30" target="_self" title="FSM_3:th.30">Th30</a></span>;<br/><span class="kw">hence </span><a NAME="E7:54_2_1_1_1_1_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:54_2_1_1_1_1_2"/>
<font color="Maroon" title="c7">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:54_2_1_1_2"/><span class="lab"><font color="Green" title="E40">A14</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">bTS</font> =  <a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font> as  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="c2">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="c2">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T31" target="_self" title="FSM_3:th.31">Th31</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">v1</font>, <font color="Maroon" title="c11">v2</font> being    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c2">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E4:54_2_1_1_2"/><span class="lab"><font color="Green" title="E41">A15</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">v1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">and </span><br/><a NAME="E5:54_2_1_1_2"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">v2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">and </span><br/><a NAME="E6:54_2_1_1_2"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c11">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E1:54_2_1_1_2"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="ref" href="fsm_3.html#T5" target="_self" title="FSM_3:th.5">Th5</a></span>;<br/><a NAME="E7:54_2_1_1_2"/><span class="lab"><font color="Green" title="E44">A18</font></span>: 
<font color="Maroon" title="c10">v1</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c10">v1</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E4:54_2_1_1_2"><span class="lab"><font color="Green" title="E41">A15</font></span></a></span>;<br/><a NAME="E8:54_2_1_1_2"/><span class="lab"><font color="Green" title="E45">A19</font></span>: 
the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Maroon" title="c9">bTS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c10">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c11">v2</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c11">v2</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c9">bTS</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite3.html#T105" title="REWRITE3:th.105">REWRITE3:105</a></span>
<br/>.= 
<font color="Maroon" title="c11">v2</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v1</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c9">bTS</font></span>)</span>,<font color="Maroon" title="c9">bTS</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T30" target="_self" title="FSM_3:th.30">Th30</a></span>
<br/>.= 
<font color="Maroon" title="c11">v2</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v1</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">bTS</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_2_1_1_2"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="rewrite3.html#T105" title="REWRITE3:th.105">REWRITE3:105</a></span>
<br/>.= 
<font color="Maroon" title="c11">v2</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v1</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_2_1_1_2"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="rewrite3.html#T105" title="REWRITE3:th.105">REWRITE3:105</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c10">v1</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c11">v2</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E5:54_2_1_1_2"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E7:54_2_1_1_2"><span class="lab"><font color="Green" title="E44">A18</font></span></a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">v1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c11">v2</font></span>)</span></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:54_2_1_1_2"/>
<font color="Maroon" title="c7">u</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_2_1_1_2"><span class="lab"><font color="Green" title="E43">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:54_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:54"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E37">A4</font></span></a>);<br/></span>
<a NAME="E5:54"/><span class="kw">then </span>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">w</font> implies <font color="Maroon" title="c3">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
;<br/>
<span class="kw">hence </span><a NAME="E6:54"/>
<font color="Maroon" title="c3">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c5">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">TS</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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