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

<span class="kw">set </span><font color="Maroon" title="c1">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">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="c2">empty1</font>,<font color="Maroon" title="c2">empty1</font> #) as   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V3" title="GRAPH_1:attr.3">non-multi</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V2" title="GRAPH_1:attr.2">oriented</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V4" title="GRAPH_1:attr.4">simple</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:10_2_1"/>
<font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#D9" target="_self" title="GRAPH_1:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V3" title="GRAPH_1:attr.3">non-multi</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V2" title="GRAPH_1:attr.2">oriented</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V4" title="GRAPH_1:attr.4">simple</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a>  )</span><br/>

<a NAME="E4:10_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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">G</font> &amp; <font color="Olive" title="b2">y</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="c3">G</font> &amp; ( (  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">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#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">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#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">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#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">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#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">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#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">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#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">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#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">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/>
<span class="kw">hence </span><a NAME="E5:10_2_1"/>
<font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V3" title="GRAPH_1:attr.3">non-multi</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#D6" target="_self" title="GRAPH_1:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V2" title="GRAPH_1:attr.2">oriented</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V4" title="GRAPH_1:attr.4">simple</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a>  )</span><br/>

<a NAME="E6:10_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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">G</font> &amp; <font color="Olive" title="b2">y</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="c3">G</font> &amp;  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">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#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">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#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">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#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">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/>
<span class="kw">hence </span><a NAME="E7:10_2_1"/>
<font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V2" title="GRAPH_1:attr.2">oriented</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#D5" target="_self" title="GRAPH_1:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V4" title="GRAPH_1:attr.4">simple</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a>  )</span><br/>

<a NAME="E8:10_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="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c3">G</font> or  not  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c3">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#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E9:10_2_1"/>
<font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V4" title="GRAPH_1:attr.4">simple</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#D7" target="_self" title="GRAPH_1: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"> <font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a> </span><br/>

<a NAME="E10:10_2_1"/>
 for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>  holds <br/>(  not  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G2</font> or  not <font color="Maroon" title="c3">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="10_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c4">G1</font>, <font color="Maroon" title="c5">G2</font> being   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a><span class="kw"> such that </span><a NAME="E2:10_2_1_1"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font>
 <span class="kw">and </span><br/><a NAME="E3:10_2_1_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c3">G</font> <a href="graph_1.html#R1" title="GRAPH_1:pred.1">is_sum_of</a> <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c5">G2</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="E4:10_2_1_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</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="#E2:10_2_1_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E5:10_2_1_1"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
(  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c4">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c5">G2</font> &amp;  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c4">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c5">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_2_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="graph_1.html#D4" target="_self" title="GRAPH_1:def.4">Def4</a></span>;<br/>
<a NAME="E6:10_2_1_1"/>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G1</font> <a href="graph_1.html#K3" title="GRAPH_1:func.3">\/</a> <font color="Maroon" title="c5">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_2_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="graph_1.html#D4" target="_self" title="GRAPH_1:def.4">Def4</a></span>;<br/>
<a NAME="E7:10_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A5</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_2_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="graph_1.html#D3" target="_self" title="GRAPH_1:def.3">Def3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">x</font> =  the   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c4">G1</font>;<br/>
<a NAME="E8:10_2_1_1"/>
(  the   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c4">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font> &amp;  the   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c4">G1</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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_2_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E9:10_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A6</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">y</font> =  the   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c5">G2</font>;<br/>
<a NAME="E10:10_2_1_1"/>
(  the   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c5">G2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font> &amp;  the   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Vertex</a> of <font color="Maroon" title="c5">G2</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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_2_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E11:10_2_1_1"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:10_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_2_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E9:10_2_1_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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">hence </span><a NAME="E11:10_2_1"/>
<font color="Maroon" title="c3">G</font> is  <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#D8" target="_self" title="GRAPH_1:def.8">Def8</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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