<?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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a></span>)</span><br/>  for <font color="Olive" title="b2">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> over <font color="Olive" title="b1">F</font> holds   <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_3.html#K3" title="FSM_3:func.3">left-Lang</a>  the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Olive" title="b2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> over <font color="Maroon" title="c2">F</font> 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#K3" title="FSM_3:func.3">left-Lang</a>  the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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> over <font color="Maroon" title="c2">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_3.html#K3" title="FSM_3:func.3">left-Lang</a>  the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c3">A</font></span><br/>





<a NAME="E1:41"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 for <font color="Olive" title="b1">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>   st <font color="Olive" title="b1">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c3">A</font> holds <br/><font color="Olive" title="b1">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K3" title="FSM_3:func.3">left-Lang</a>  the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c3">A</font> implies <font color="Maroon" title="c4">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K3" title="FSM_3:func.3">left-Lang</a>  the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c3">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1"/>
<font color="Maroon" title="c4">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K3" title="FSM_3:func.3">left-Lang</a>  the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c3">A</font></span><br/>

<a NAME="E2:41_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="c3">A</font>,<font color="Maroon" title="c3">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="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T19" target="_self" onmouseover="rs('fsm_3/T19')" onmouseout="rh()">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:41_1"/>
<font color="Maroon" title="c4">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K3" title="FSM_3:func.3">left-Lang</a>  the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:41"/>
 for <font color="Olive" title="b1">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>   st <font color="Olive" title="b1">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K3" title="FSM_3:func.3">left-Lang</a>  the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c3">A</font> holds <br/><font color="Olive" title="b1">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c3">A</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K3" title="FSM_3:func.3">left-Lang</a>  the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c3">A</font> implies <font color="Maroon" title="c4">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c3">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_2"/>
<font color="Maroon" title="c4">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K3" title="FSM_3:func.3">left-Lang</a>  the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c3">A</font></span><br/>

<a NAME="E2:41_2"/><span class="kw">then </span>
 the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c3">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</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="c3">A</font>,<font color="Maroon" title="c3">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T15" target="_self" onmouseover="rs('fsm_3/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:41_2"/>
<font color="Maroon" title="c4">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T19" target="_self" onmouseover="rs('fsm_3/T19')" onmouseout="rh()">Th19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:41"/>
 <a href="fsm_3.html#K6" title="FSM_3:func.6">Lang</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_3.html#K3" title="FSM_3:func.3">left-Lang</a>  the <a href="fsm_3.html#U2" title="FSM_3:sel.2">FinalS</a> of <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="subset_1.html#T3" onmouseover="rs('subset_1/T3')" onmouseout="rh()">SUBSET_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
