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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</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><br/> for <font color="Olive" title="b2">U</font>, <font color="Olive" title="b3">V</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">v1</font>, <font color="Olive" title="b5">v2</font> being  <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">U</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b3">V</font> &amp; <font color="Olive" title="b4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font> &amp; ( for <font color="Olive" title="b6">v3</font>, <font color="Olive" title="b7">v4</font> being  <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b6">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font> &amp; <font color="Olive" title="b7">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">V</font> holds <br/>for <font color="Olive" title="b8">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b8">e</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> or  not <font color="Olive" title="b8">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b6">v3</font>,<font color="Olive" title="b7">v4</font> ) ) &amp; <font color="Olive" title="b1">p</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b5">v2</font> holds <br/><font color="Olive" title="b1">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b5">v2</font>,<font color="Olive" title="b2">U</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">U</font>, <font color="Olive" title="b2">V</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being  <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">U</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b3">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; ( for <font color="Olive" title="b5">v3</font>, <font color="Olive" title="b6">v4</font> being  <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b5">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b6">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> holds <br/>for <font color="Olive" title="b7">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b7">e</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> or  not <font color="Olive" title="b7">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b5">v3</font>,<font color="Olive" title="b6">v4</font> ) ) &amp; <font color="Maroon" title="c2">p</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Olive" title="b1">U</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">U</font>, <font color="Maroon" title="c4">V</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">v1</font>, <font color="Olive" title="b2">v2</font> being  <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">U</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">V</font> &amp; <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> &amp; ( for <font color="Olive" title="b3">v3</font>, <font color="Olive" title="b4">v4</font> being  <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b3">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b4">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> holds <br/>for <font color="Olive" title="b5">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b5">e</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> or  not <font color="Olive" title="b5">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b3">v3</font>,<font color="Olive" title="b4">v4</font> ) ) &amp; <font color="Maroon" title="c2">p</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Maroon" title="c3">U</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">v2</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">U</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">V</font> &amp; <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> &amp; ( for <font color="Olive" title="b1">v3</font>, <font color="Olive" title="b2">v4</font> being  <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b2">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> holds <br/>for <font color="Olive" title="b3">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b3">e</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> or  not <font color="Olive" title="b3">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b1">v3</font>,<font color="Olive" title="b2">v4</font> ) ) &amp; <font color="Maroon" title="c2">p</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font> implies <font color="Maroon" title="c2">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c3">U</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">U</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">V</font>
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">and </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
for <font color="Olive" title="b1">v3</font>, <font color="Olive" title="b2">v4</font> being  <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b2">v4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> holds <br/>for <font color="Olive" title="b3">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b3">e</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> or  not <font color="Olive" title="b3">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Olive" title="b1">v3</font>,<font color="Olive" title="b2">v4</font> )
 <span class="kw">and </span><br/><a NAME="E4:16"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="graph_5.html#R1" title="GRAPH_5:pred.1">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</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="c2">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c3">U</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">FS</font> = the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">FT</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font>;<br/>
<div><span class="lab"><font color="Green" title="E21">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1"/>
not  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c10">q</font>, <font color="Maroon" title="c11">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 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"> such that </span><br/><a NAME="E3:16_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c2">p</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
not  <a href="graph_5.html#K1" title="GRAPH_5:func.1">vertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">and </span><br/><a NAME="E5:16_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">and </span><br/><a NAME="E6:16_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">r</font>
 <span class="kw">and </span><br/><a NAME="E7:16_1"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
 <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c10">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#T23" title="GRAPH_5:th.23">GRAPH_5:23</a></span>;<br/><a NAME="E8:16_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="#E3:16_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="graphsp.html#T2" target="_self" title="GRAPHSP:th.2">Th2</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E9:16_1"/>
<font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E10:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A12</font></span>: 
 <a href="graph_5.html#K1" title="GRAPH_5:func.1">vertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">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="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D1" title="GRAPH_5:def.1">GRAPH_5:def 1</a></span>;<br/><div><span class="lab"><font color="Green" title="E29">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1_1"/>
( 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="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> &amp; 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="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:16_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="graph_5.html#K1" title="GRAPH_5:func.1">vertices</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c3">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T48" title="ZFMISC_1:th.48">ZFMISC_1:48</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>; <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><a NAME="E12:16_1"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">vy</font> = 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="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="graphsp.html#E13"><span class="lab"><font color="Green" title="E13">Lm3</font></span></a></span>;<br/><a NAME="E14:16_1"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
( <font color="Maroon" title="c12">vy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> or <font color="Maroon" title="c12">vy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1_2"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_2_1"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_1_2_1"/><span class="kw">then </span>
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="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></span>;<br/><a NAME="E3:16_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c12">vy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:16_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E11:16_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E8:16_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E14:16_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E1:16_1_2_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="graph_5.html#D3" title="GRAPH_5:def.3">GRAPH_5:def 3</a></span>; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_2_2"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c13">vx</font> = 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="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E12:16_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="graphsp.html#E13"><span class="lab"><font color="Green" title="E13">Lm3</font></span></a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1_2_2_1_1"/>
( <font color="Maroon" title="c13">vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> or  not <font color="Maroon" title="c13">vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E33">A18</font></span>: 
<font color="Maroon" title="c13">vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_1_2_2_1_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c13">vx</font>,<font color="Maroon" title="c12">vy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_4.html#D1" title="GRAPH_4:def.1">GRAPH_4:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_1_2_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E12:16_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E11:16_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E14:16_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E1:16_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="graphsp.html#T2" target="_self" title="GRAPHSP:th.2">Th2</a></span>; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E33">A19</font></span>: 
not <font color="Maroon" title="c13">vx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:16_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E35">A21</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_2_2"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">vq</font> = 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="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> as   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="graphsp.html#E13"><span class="lab"><font color="Green" title="E13">Lm3</font></span></a></span>;<br/><a NAME="E5:16_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
<font color="Maroon" title="c14">vq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c10">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E3:16_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="graphsp.html#E15"><span class="lab"><font color="Green" title="E15">Lm4</font></span></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="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</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="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="txt" href="#E6:16_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E3:16_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="graphsp.html#E6"><span class="lab"><font color="Green" title="E6">Lm1</font></span></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="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="#E2:16_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="graph_1.html#D13" title="GRAPH_1:def.13">GRAPH_1:def 13</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:16_1_2_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E1:16_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E5:16_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E36">A22</font></span></a></span>; <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></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E6:16"/>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E7:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="graph_5.html#K2" title="GRAPH_5:func.2">vertices</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">v2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:16"/>
<font color="Maroon" title="c2">p</font> <a href="graph_5.html#R2" title="GRAPH_5:pred.2">is_orientedpath_of</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="graph_5.html#D4" title="GRAPH_5:def.4">GRAPH_5:def 4</a></span>; <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>
