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<span class="kw">let </span><font color="Maroon" title="c1">E</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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> <br/> for <font color="Olive" title="b2">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="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span><br/> for <font color="Olive" title="b3">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 <font color="Olive" title="b2">F</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> <font color="Olive" title="b3">A</font> iff <font color="Olive" title="b1">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="Olive" title="b3">A</font>,<font color="Olive" title="b3">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="Olive" title="b3">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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"> 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="catalan2.html#K2" title="CATALAN2:func.2">^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> of <font color="Olive" title="b1">F</font> holds <br/> ( <font color="Maroon" title="c2">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="Olive" title="b2">A</font> iff <font color="Maroon" title="c2">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="Olive" title="b2">A</font>,<font color="Olive" title="b2">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="Olive" title="b2">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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 <font color="Maroon" title="c3">F</font> holds <br/> ( <font color="Maroon" title="c2">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="Olive" title="b1">A</font> iff <font color="Maroon" title="c2">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="Olive" title="b1">A</font>,<font color="Olive" title="b1">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="Olive" title="b1">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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 <font color="Maroon" title="c3">F</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="c2">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="c4">A</font> iff <font color="Maroon" title="c2">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="c4">A</font>,<font color="Maroon" title="c4">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="c4">A</font> )</span><br/>







<span class="kw">thus </span><a NAME="E1:41"/>
( <font color="Maroon" title="c2">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="c4">A</font> implies <font color="Maroon" title="c2">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="c4">A</font>,<font color="Maroon" title="c4">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="c4">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="c2">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="c4">A</font>,<font color="Maroon" title="c4">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="c4">A</font> implies <font color="Maroon" title="c2">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="c4">A</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:41_1"/>
<font color="Maroon" title="c2">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="c4">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="c2">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="c4">A</font>,<font color="Maroon" title="c4">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="c4">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">A</font><span class="kw"> such that </span><br/><a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E4:41_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c6">q</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="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E5:41_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">w</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#T19" target="_self" title="FSM_3:th.19">Th19</a></span>;<br/>
<a NAME="E6:41_1"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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="c4">A</font>,<font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E5:41_1"><span class="lab"><font color="Green" title="E26">A3</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="E7:41_1"/>
<font color="Maroon" title="c2">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="c4">A</font>,<font color="Maroon" title="c4">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="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</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">assume </span><a NAME="E2:41"/>
<font color="Maroon" title="c2">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="c4">A</font>,<font color="Maroon" title="c4">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="c4">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="c2">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="c4">A</font></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="E4:41"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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="c4">A</font>,<font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E5:41"/><span class="lab"><font color="Green" title="E25">A5</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 <font color="Maroon" title="c4">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E24">A4</font></span></a></span>;<br/>
<a NAME="E7:41"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">A</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_3.html#U1" title="FSM_3:sel.1">InitS</a> of <font color="Maroon" title="c4">A</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">w</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E24">A4</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="E8:41"/>
<font color="Maroon" title="c2">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="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E25">A5</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>
