<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="funct_2.html#NM1">Function</a> of  <a href="xboole_0.html#K1">{}</a> ,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T55">FUNCT_2:55</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="graph_1.html#G1" target="_self">MultiGraphStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) as   <a href="graph_1.html#NM1">Graph</a> <b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E3:11_2_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="graph_1.html#V7" target="_self">finite</a>
 <b>by </b><i><a class="ref" href="graph_1.html#D8" target="_self">Def8</a></i>;<br/>

<a NAME="E4:11_2_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>5</sub></font> &amp; ( ( the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) or ( the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
;<br/>
<b>hence </b><a NAME="E5:11_2_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="graph_1.html#V4" target="_self">non-multi</a>
 <b>by </b><i><a class="ref" href="graph_1.html#D5" target="_self">Def5</a></i>;<br/>

<a NAME="E6:11_2_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>5</sub></font> &amp; the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
;<br/>
<b>hence </b><a NAME="E7:11_2_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="graph_1.html#V3" target="_self">oriented</a>
 <b>by </b><i><a class="ref" href="graph_1.html#D4" target="_self">Def4</a></i>;<br/>

<a NAME="E8:11_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U2" target="_self">Edges</a> of <font color="Maroon">c<sub>5</sub></font> or  not the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E9:11_2_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="graph_1.html#V5" target="_self">simple</a>
 <b>by </b><i><a class="ref" href="graph_1.html#D6" target="_self">Def6</a></i>;<br/>

<a NAME="E10:11_2_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="graph_1.html#NM1">Graph</a> holds <br/>(  not the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Olive">b<sub>2</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="graph_1.html#R1" target="_self">is_sum_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="graph_1.html#NM1">Graph</a><b> such that </b><a NAME="E1:11_2_1_1"/><i><font color="Green">E11</font></i>: 
( the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="graph_1.html#R1" target="_self">is_sum_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<a NAME="E2:11_2_1_1"/><i><font color="Green">E12</font></i>: 
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:11_2_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E3:11_2_1_1"/>
( the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>7</sub></font> &amp; the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#R1">tolerates</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="graph_1.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:11_2_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="graph_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:11_2_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E6:11_2_1_1"/><i><font color="Green">E14</font></i>: 
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>6</sub></font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> 
 
<b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E7:11_2_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<a NAME="E8:11_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E4:11_2_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="graph_1.html#E6:11_2_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:11_2_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
1 <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E7:11_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E11:11_2_1_1"/><i><font color="Green">E17</font></i>: 
the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>7</sub></font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> 
 
<b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:11_2_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<a NAME="E13:11_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E4:11_2_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="graph_1.html#E11:11_2_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E14:11_2_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a> the <a href="graph_1.html#U1" target="_self">Vertices</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E12:11_2_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:11_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_1.html#E2:11_2_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="graph_1.html#E9:11_2_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:11_2_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="graph_1.html#V6" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="graph_1.html#D7" target="_self">Def7</a></i>;<br/>


</div>
