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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a>   <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</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">v</font> being   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font>  holds  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Olive" title="b1">v</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/><span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</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_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c2">v</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/>



<span class="kw">assume </span><a NAME="E1:47"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
 <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c2">v</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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">E</font> =  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>;<br/>
<span class="lab"><font color="Green" title="E38">A2</font></span>:  <a href="graph_1.html#K8" title="GRAPH_1:func.8">Degree</a> <font color="Maroon" title="c2">v</font> = 
 <a href="graph_3.html#K7" title="GRAPH_3:func.7">Degree</a> (<font color="Maroon" title="c2">v</font>, 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="ref" href="graph_3.html#T24" target="_self" title="GRAPH_3:th.24">Th29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> (<font color="Maroon" title="c2">v</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>)</span>)</span></span>)</span>

;<br/>
<a NAME="E3:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A3</font></span>: 
 <a href="graph_3.html#K2" title="GRAPH_3:func.2">Edges_In</a> (<font color="Maroon" title="c2">v</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E37">A1</font></span></a></span>;<br/>
<a NAME="E4:47"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
 <a href="graph_3.html#K3" title="GRAPH_3:func.3">Edges_Out</a> (<font color="Maroon" title="c2">v</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E38">A2</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S</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="c5">T</font> =  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:47"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
<font color="Maroon" title="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">s</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> <font color="Maroon" title="c6">e</font> as   <a href="graph_3.html#NM2" title="GRAPH_3:NM.2">Vertex</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_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="E1:47_2"/>
( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> or <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1"><span class="kw">suppose </span></a><a NAME="E1:47_2_1"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:47_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="graph_3.html#D2" target="_self" title="GRAPH_3:def.2">Def2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_2"><span class="kw">suppose </span></a><a NAME="E1:47_2_2"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">c</font> being    <a href="graph_1.html#M1" title="GRAPH_1:mode.1">Chain</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c9">vs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:47_2_2"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
 not <font color="Maroon" title="c8">c</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">and </span><br/><a NAME="E4:47_2_2"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<font color="Maroon" title="c9">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">and </span><br/><a NAME="E5:47_2_2"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
<font color="Maroon" title="c9">vs</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="c2">v</font>
 <span class="kw">and </span><br/><a NAME="E6:47_2_2"/>
<font color="Maroon" title="c9">vs</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="c9">vs</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_3.html#T18" target="_self" title="GRAPH_3:th.18">Th23</a></span>;<br/><a NAME="E7:47_2_2"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_2_2"><span class="lab"><font color="Green" title="E42">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="E8:47_2_2"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E9:47_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A10</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E10:47_2_2"/>
<font color="Maroon" title="c8">c</font> is    <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">by</span> <span class="lab"><a class="ref" href="msscyc_1.html#D1" title="MSSCYC_1:def.1">MSSCYC_1:def 1</a></span>;<br/><a NAME="E11:47_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A11</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E12:47_2_2"/>
( <font color="Maroon" title="c9">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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="c8">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> or <font color="Maroon" title="c9">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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="c8">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_2_2"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E7:47_2_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="graph_3.html#E16"><span class="lab"><font color="Green" title="E15">Lm3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E13:47_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E5:47_2_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E11:47_2_2"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="txt" href="#E9:47_2_2"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="ref" href="graph_3.html#D1" target="_self" title="GRAPH_3:def.1">Def1</a>, <a class="ref" href="graph_3.html#D2" target="_self" title="GRAPH_3:def.2">Def2</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></div><span class="kw">end;</span></div>

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