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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">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><br/>  for <font color="Olive" title="b3">W</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>,<a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> holds <br/> (  <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Olive" title="b3">W</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff  ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">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> &amp; <font color="Olive" title="b4">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</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="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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>,<a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> holds <br/> (  <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Olive" title="b1">W</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff  ex <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">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> &amp; <font color="Olive" title="b2">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</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">W</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>,<a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">W</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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> &amp; <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> ) )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">EG</font> =  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">G</font>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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> &amp; <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> ) implies  <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">W</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">W</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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> &amp; <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> )</span><br/><span class="kw">assume </span><a NAME="E2:41_1"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">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="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</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"> contradiction</span><br/><a NAME="E3:41_1"/><span class="kw">then </span>
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graphsp.html#D7" target="_self" title="GRAPHSP:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:41_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E34">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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:41"/>
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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> &amp; <font color="Olive" title="b1">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</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">  <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">W</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">then </span><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="E4:41"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <a href="graphsp.html#K10" title="GRAPHSP:func.10">XEdge</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e</font>
 <span class="kw">and </span><br/><a NAME="E5:41"/><span class="lab"><font color="Green" title="E35">A3</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">and </span><br/><a NAME="E6:41"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="graph_4.html#R1" title="GRAPH_4:pred.1">orientedly_joins</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="ref" href="graphsp.html#D6" target="_self" title="GRAPHSP:def.6">Def6</a></span>;<br/>
<a NAME="E7:41"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E8:41"/><span class="kw">then </span>
<font color="Maroon" title="c4">W</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T4" title="PARTFUN1:th.4">PARTFUN1:4</a></span>;<br/>
<a NAME="E9:41"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c4">W</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D12" title="GRAPH_5:def.12">GRAPH_5:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:41"/>
 <a href="graphsp.html#K12" title="GRAPHSP:func.12">Weight</a> (<font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">W</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="graphsp.html#D7" target="_self" title="GRAPHSP:def.7">Def7</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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