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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">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">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="Olive" title="b1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span><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 <font color="Olive" title="b2">F</font><br/> for <font color="Olive" title="b4">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b3">TS</font> holds <br/> ( <font color="Olive" title="b4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c2">X</font>,<font color="Olive" title="b3">TS</font> iff  ex <font color="Olive" title="b5">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b3">TS</font> st <br/>( <font color="Olive" title="b5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b5">t</font>,<font color="Maroon" title="c1">x</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b4">s</font>,<font color="Olive" title="b3">TS</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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">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="c3">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span><br/> for <font color="Olive" title="b2">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 <font color="Olive" title="b1">F</font><br/> for <font color="Olive" title="b3">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">TS</font> holds <br/> ( <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c2">X</font>,<font color="Olive" title="b2">TS</font> iff  ex <font color="Olive" title="b4">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">TS</font> st <br/>( <font color="Olive" title="b4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b4">t</font>,<font color="Maroon" title="c1">x</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b3">s</font>,<font color="Olive" title="b2">TS</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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="c3">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">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 <font color="Maroon" title="c4">F</font><br/> for <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">TS</font> holds <br/> ( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c2">X</font>,<font color="Olive" title="b1">TS</font> iff  ex <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">TS</font> st <br/>( <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b3">t</font>,<font color="Maroon" title="c1">x</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">TS</font> ) )</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 <font color="Maroon" title="c4">F</font>; <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">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">TS</font> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c5">TS</font> iff  ex <font color="Olive" title="b2">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">TS</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">t</font>,<font color="Maroon" title="c1">x</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c5">TS</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">TS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c5">TS</font> iff  ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">TS</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c1">x</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">TS</font> ) )</span><br/>











<span class="kw">thus </span><a NAME="E1:113"/>
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c5">TS</font> implies  ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">TS</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c1">x</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">TS</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">TS</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c1">x</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">TS</font> ) implies <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c5">TS</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:113_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c5">TS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">TS</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c1">x</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">TS</font> )</span><br/>

<a NAME="E2:113_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">TS</font> st <br/>( <font color="Olive" title="b1">s9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> &amp;  ex <font color="Olive" title="b2">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">TS</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">t</font>,<font color="Maroon" title="c1">x</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Olive" title="b1">s9</font>,<font color="Maroon" title="c5">TS</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:113_1"/>
 ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">TS</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c1">x</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">TS</font> )
 ; <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">thus </span><a NAME="E2:113"/>
(  ex <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">TS</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b1">t</font>,<font color="Maroon" title="c1">x</font> <a href="rewrite3.html#R4" title="REWRITE3:pred.4">==&gt;*</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c5">TS</font> ) implies <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c5">TS</font> )
 ; <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>
