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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds   <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> (<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_5.html#K3" title="GRAPH_5:func.3">OrientedPaths</a> (<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_5.html#K3" title="GRAPH_5:func.3">OrientedPaths</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>) or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K3" title="GRAPH_5:func.3">OrientedPaths</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>) )</span><br/>







<span class="kw">assume </span><a NAME="E1:50"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K4" title="GRAPH_5:func.4">AcyclicPaths</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K3" title="GRAPH_5:func.3">OrientedPaths</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">p</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_4.html#V1" title="GRAPH_4:attr.1">Simple</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">and </span><br/><a NAME="E4:50"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="graph_5.html#R3" title="GRAPH_5:pred.3">is_acyclicpath_of</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>
 ;<br/>
<a NAME="E5:50"/>
<font color="Maroon" title="c5">p</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="graph_5.html#D6" target="_self" title="GRAPH_5:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:50"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K3" title="GRAPH_5:func.3">OrientedPaths</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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