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

<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">A</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_3.html#L2" title="FSM_3:struct.2">automaton</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> holds   <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Olive" title="b1">A</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_3.html#L2" title="FSM_3:struct.2">automaton</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">  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">DA</font> =  <a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font>;<br/>
<a NAME="E1:61"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
for <font color="Olive" title="b1">w</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">w</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"> ( <font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1"/>
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c2">A</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="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span></span><br/>

<a NAME="E2:61_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T20" target="_self" title="FSM_3:th.20">Th20</a></span>;<br/>
<a NAME="E3:61_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E4:61_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T34" target="_self" title="FSM_3:th.34">Th34</a></span>;<br/>
<a NAME="E5:61_1"/>
<font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</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"><font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</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#T38" target="_self" title="FSM_3:th.38">Th38</a></span>;<br/>
<a NAME="E6:61_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E7:61_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:61_1"/>
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T20" target="_self" title="FSM_3:th.20">Th20</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>
<a NAME="E2:61"/>
for <font color="Olive" title="b1">w</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span> holds <br/><font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">w</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"> ( <font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span> implies <font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_2"/>
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</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="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c2">A</font></span><br/>

<a NAME="E2:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T20" target="_self" title="FSM_3:th.20">Th20</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:61_2"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:61_2"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E6:61_2"/>
<font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</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"><font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</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#T38" target="_self" title="FSM_3:th.38">Th38</a></span>;<br/>
<a NAME="E7:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_2"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:61_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">w</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_2"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="fsm_3.html#T35" target="_self" title="FSM_3:th.35">Th35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:61_2"/>
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T20" target="_self" title="FSM_3:th.20">Th20</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="E3:61"/>
 <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <span class="p1">(<span class="default"><a href="fsm_3.html#K4" title="FSM_3:func.4">_bool</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</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>
