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<b>let </b><font color="Maroon" title="c1">X</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> ex <font color="Olive" title="b2">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> st <font color="Olive" title="b2">g</font> <a href="jgraph_1.html#R1" title="JGRAPH_1:pred.1">is_Shortcut_of</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies  ex <font color="Olive" title="b1">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> st <font color="Olive" title="b1">g</font> <a href="jgraph_1.html#R1" title="JGRAPH_1:pred.1">is_Shortcut_of</a> <font color="Maroon" title="c2">f</font> )</span><br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E10">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> st <font color="Olive" title="b1">g</font> <a href="jgraph_1.html#R1" title="JGRAPH_1:pred.1">is_Shortcut_of</a> <font color="Maroon" title="c2">f</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c3">f1</font> = <font color="Maroon" title="c2">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" title="JGRAPH_1:func.1">PGraph</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<a NAME="E3:13"/>
<font color="Maroon" title="c3">f1</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a>  <a href="jgraph_1.html#K4" title="JGRAPH_1:func.4">PairF</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:13"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="jgraph_1.html#T11" target="_self" title="JGRAPH_1:th.11">Th11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">fc</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K4" title="JGRAPH_1:func.4">PairF</a> <font color="Maroon" title="c2">f</font></span>)</span>, <font color="Maroon" title="c5">fvs</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c6">sc</font> being  <a href="graph_1.html#V8" title="GRAPH_1:attr.8">oriented</a> <a href="graph_2.html#V3" title="GRAPH_2:attr.3">simple</a>  <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of  <a href="jgraph_1.html#K1" title="JGRAPH_1:func.1">PGraph</a> <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c7">vs1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" title="JGRAPH_1:func.1">PGraph</a> <font color="Maroon" title="c1">X</font></span>)</span><b> such that </b><br/><a NAME="E5:13"/><i><font color="Green" title="E11">A2</font></i>: 
(  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c4">fc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">sc</font> &amp;  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c5">fvs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">vs1</font> &amp; <font color="Maroon" title="c7">vs1</font> <a href="graph_4.html#R3" title="GRAPH_4:pred.3">is_oriented_vertex_seq_of</a> <font color="Maroon" title="c6">sc</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">vs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">vs1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="graph_4.html#T23" title="GRAPH_4:th.23">GRAPH_4:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">g1</font> = <font color="Maroon" title="c7">vs1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E7:13"/>
<font color="Maroon" title="c8">g1</font> <a href="jgraph_1.html#R1" title="JGRAPH_1:pred.1">is_Shortcut_of</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:13"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="jgraph_1.html#D3" target="_self" title="JGRAPH_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E8:13"/>
 ex <font color="Olive" title="b1">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> st <font color="Olive" title="b1">g</font> <a href="jgraph_1.html#R1" title="JGRAPH_1:pred.1">is_Shortcut_of</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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