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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#V2" title="GRAPH_1:attr.2">oriented</a>  <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">vs</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><br/>  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">p</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b1">vs</font> &amp; <font color="Olive" title="b3">q</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Olive" title="b1">vs</font> holds <br/><font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">vs</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">p</font>, <font color="Olive" title="b2">q</font> being   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">p</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c2">vs</font> &amp; <font color="Olive" title="b2">q</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c2">vs</font> holds <br/><font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be   <a href="graph_1.html#V7" title="GRAPH_1:attr.7">oriented</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</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"> ( <font color="Maroon" title="c3">p</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c2">vs</font> &amp; <font color="Maroon" title="c4">q</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c2">vs</font> implies <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:85"/><span class="lab"><font color="Green" title="E71">A1</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c2">vs</font>
 <span class="kw">and </span><br/><a NAME="E2:85"/><span class="lab"><font color="Green" title="E72">A2</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="graphsp.html#R5" title="GRAPHSP:pred.5">is_oriented_edge_seq_of</a> <font color="Maroon" title="c2">vs</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="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font></span><br/>

<span class="lab"><font color="Green" title="E73">A3</font></span>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">vs</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> implies <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:85_2"/><span class="lab"><font color="Green" title="E74">A4</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</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="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A5</font></span>:  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span> = 
<font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></span>
<br/>.= 
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="txt" href="#E1:85_2"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></span>
;<br/><a NAME="E3:85_2"/><span class="lab"><font color="Green" title="E76">A6</font></span>: 
( <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="txt" href="#E1:85_2"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="ref" href="graphsp.html#T2" target="_self" title="GRAPHSP:th.2">Th2</a></span>;<br/> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span> = 
<font color="Maroon" title="c2">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="txt" href="#E1:85_2"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></span>
<br/>.= 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="txt" href="#E1:85_2"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="ref" href="graphsp.html#D19" target="_self" title="GRAPHSP:def.19">Def19</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:85_2"/>
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85_2"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="txt" href="#E3:85_2"><span class="lab"><font color="Green" title="E76">A6</font></span></a>, <a class="ref" href="graph_1.html#D5" title="GRAPH_1:def.5">GRAPH_1:def 5</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:85"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</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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