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

<b>set </b><font color="Maroon" title="c3">V</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">empty1</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T55" title="FUNCT_2:th.55">FUNCT_2:55</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">G</font> =  <a href="graph_1.html#G1" title="GRAPH_1:aggr.1">MultiGraphStruct</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c4">empty1</font>,<font color="Maroon" title="c4">empty1</font> #) as   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a> <b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self" title="GRAPH_1:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">G</font>
;<br/>

<b>thus </b><a NAME="E3:11_2_1"/>
<font color="Maroon" title="c5">G</font> is <a href="graph_1.html#V7" title="GRAPH_1:attr.7">finite</a>
 <b>by </b><i><a class="ref" href="graph_1.html#D8" target="_self" title="GRAPH_1:def.8">Def8</a></i>;<br/>

<a NAME="E4:11_2_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c5">G</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c5">G</font> &amp; ( ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> ) or ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) ) holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
;<br/>
<b>hence </b><a NAME="E5:11_2_1"/>
<font color="Maroon" title="c5">G</font> is <a href="graph_1.html#V4" title="GRAPH_1:attr.4">non-multi</a>
 <b>by </b><i><a class="ref" href="graph_1.html#D5" target="_self" title="GRAPH_1:def.5">Def5</a></i>;<br/>

<a NAME="E6:11_2_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c5">G</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c5">G</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
;<br/>
<b>hence </b><a NAME="E7:11_2_1"/>
<font color="Maroon" title="c5">G</font> is <a href="graph_1.html#V3" title="GRAPH_1:attr.3">oriented</a>
 <b>by </b><i><a class="ref" href="graph_1.html#D4" target="_self" title="GRAPH_1:def.4">Def4</a></i>;<br/>

<a NAME="E8:11_2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c5">G</font> or  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 
;<br/>
<b>hence </b><a NAME="E9:11_2_1"/>
<font color="Maroon" title="c5">G</font> is <a href="graph_1.html#V5" title="GRAPH_1:attr.5">simple</a>
 <b>by </b><i><a class="ref" href="graph_1.html#D6" target="_self" title="GRAPH_1:def.6">Def6</a></i>;<br/>

<a NAME="E10:11_2_1"/>
for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being  <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>  holds <br/>(  not the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Olive" title="b1">G1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Olive" title="b2">G2</font> or  not <font color="Maroon" title="c5">G</font> <a href="graph_1.html#R1" title="GRAPH_1:pred.1">is_sum_of</a> <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</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" title="c6">G1</font>, <font color="Maroon" title="c7">G2</font> being   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a><b> such that </b><a NAME="E1:11_2_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c6">G1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c7">G2</font> &amp; <font color="Maroon" title="c5">G</font> <a href="graph_1.html#R1" title="GRAPH_1:pred.1">is_sum_of</a> <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font> )
 ;<br/>

<a NAME="E2:11_2_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c6">G1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c7">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:11_2_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E3:11_2_1_1"/>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c6">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c7">G2</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c6">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c7">G2</font> &amp; <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c7">G2</font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:11_2_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></i>;<br/>
<a NAME="E4:11_2_1_1"/><b>then </b><i><font color="Green" title="E13">A3</font></i>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c6">G1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c7">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="graph_1.html#D2" target="_self" title="GRAPH_1:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c6">G1</font>;<br/>
<a NAME="E6:11_2_1_1"/><i><font color="Green" title="E14">A4</font></i>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c6">G1</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 
<b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self" title="GRAPH_1:def.1">Def1</a></i>;<br/>
<a NAME="E7:11_2_1_1"/><b>then </b><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c6">G1</font>
 
;<br/>
<a NAME="E8:11_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E4:11_2_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="graph_1.html#E6:11_2_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:11_2_1_1"/><b>then </b><i><font color="Green" title="E16">A6</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c6">G1</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E7:11_2_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c7">G2</font>;<br/>
<a NAME="E11:11_2_1_1"/><i><font color="Green" title="E17">A7</font></i>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c7">G2</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 
<b>by </b><i><a class="ref" href="graph_1.html#D1" target="_self" title="GRAPH_1:def.1">Def1</a></i>;<br/>
<a NAME="E12:11_2_1_1"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c7">G2</font>
 
;<br/>
<a NAME="E13:11_2_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="graph_1.html#E4:11_2_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="graph_1.html#E11:11_2_1_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E14:11_2_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c7">G2</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E12:11_2_1_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="E12">A2</font></i></a>, <a class="txt" href="graph_1.html#E9:11_2_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:11_2_1"/>
<font color="Maroon" title="c5">G</font> is <a href="graph_1.html#V6" title="GRAPH_1:attr.6">connected</a>
 <b>by </b><i><a class="ref" href="graph_1.html#D7" target="_self" title="GRAPH_1:def.7">Def7</a></i>;<br/>


</div>
