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<span class="kw">let </span><font color="Maroon" title="c1">PTN</font> be   <a href="petri.html#NM2" title="PETRI:NM.2">Petri_net</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">T0</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">PTN</font><br/>  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="petri.html#K8" title="PETRI:func.8">*'</a> <font color="Olive" title="b1">T0</font> iff  ex <font color="Olive" title="b3">f</font> being   <a href="petri.html#NM7" title="PETRI:NM.7">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b4">t</font> being   <a href="petri.html#NM5" title="PETRI:NM.5">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b4">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">T0</font> &amp; <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b4">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">T0</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">PTN</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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="petri.html#K8" title="PETRI:func.8">*'</a> <font color="Maroon" title="c2">T0</font> iff  ex <font color="Olive" title="b2">f</font> being   <a href="petri.html#NM7" title="PETRI:NM.7">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b3">t</font> being   <a href="petri.html#NM5" title="PETRI:NM.5">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <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"> ( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="petri.html#K8" title="PETRI:func.8">*'</a> <font color="Maroon" title="c2">T0</font> iff  ex <font color="Olive" title="b1">f</font> being   <a href="petri.html#NM7" title="PETRI:NM.7">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b2">t</font> being   <a href="petri.html#NM5" title="PETRI:NM.5">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b2">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>





<span class="kw">thus </span><a NAME="E1:20"/>
( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="petri.html#K8" title="PETRI:func.8">*'</a> <font color="Maroon" title="c2">T0</font> implies  ex <font color="Olive" title="b1">f</font> being   <a href="petri.html#NM7" title="PETRI:NM.7">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b2">t</font> being   <a href="petri.html#NM5" title="PETRI:NM.5">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b2">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"> (  ex <font color="Olive" title="b1">f</font> being   <a href="petri.html#NM7" title="PETRI:NM.7">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b2">t</font> being   <a href="petri.html#NM5" title="PETRI:NM.5">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b2">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) implies <font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="petri.html#K8" title="PETRI:func.8">*'</a> <font color="Maroon" title="c2">T0</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="petri.html#K8" title="PETRI:func.8">*'</a> <font color="Maroon" title="c2">T0</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">  ex <font color="Olive" title="b1">f</font> being   <a href="petri.html#NM7" title="PETRI:NM.7">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b2">t</font> being   <a href="petri.html#NM5" title="PETRI:NM.5">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b2">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">s</font> being   <a href="petri.html#NM3" title="PETRI:NM.3">place</a> of <font color="Maroon" title="c1">PTN</font><span class="kw"> such that </span><br/><a NAME="E3:20_1"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font>
 <span class="kw">and </span><br/><a NAME="E4:20_1"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 ex <font color="Olive" title="b1">f</font> being   <a href="petri.html#NM7" title="PETRI:NM.7">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font> ex <font color="Olive" title="b2">t</font> being   <a href="petri.html#NM5" title="PETRI:NM.5">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Olive" title="b2">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="petri.html#NM7" title="PETRI:NM.7">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font>, <font color="Maroon" title="c6">t</font> being   <a href="petri.html#NM5" title="PETRI:NM.5">transition</a> of <font color="Maroon" title="c1">PTN</font><span class="kw"> such that </span><br/><a NAME="E6:20_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
( <font color="Maroon" title="c6">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">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">  ex <font color="Olive" title="b1">t</font> being   <a href="petri.html#NM5" title="PETRI:NM.5">transition</a> of <font color="Maroon" title="c1">PTN</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">t</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="c6">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E7:20_1"/>
( <font color="Maroon" title="c6">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T0</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E6:20_1"><span class="lab"><font color="Green" title="E8">A3</font></span></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">given </span><font color="Maroon" title="c4">f</font> being   <a href="petri.html#NM7" title="PETRI:NM.7">S-T_arc</a> of <font color="Maroon" title="c1">PTN</font>, <font color="Maroon" title="c5">t</font> being   <a href="petri.html#NM5" title="PETRI:NM.5">transition</a> of <font color="Maroon" title="c1">PTN</font><span class="kw"> such that </span><a NAME="E3:20"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">T0</font>
 <span class="kw">and </span><br/><a NAME="E4:20"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="petri.html#K8" title="PETRI:func.8">*'</a> <font color="Maroon" title="c2">T0</font></span><br/>

<a NAME="E5:20"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="petri.html#K2" title="PETRI:func.2">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:20"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="petri.html#K8" title="PETRI:func.8">*'</a> <font color="Maroon" title="c2">T0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E7">A5</font></span></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>
