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

<span class="kw">let </span><font color="Maroon" title="c1">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">E</font>  st <font color="Olive" title="b1">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c1">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  &amp; (  for <font color="Olive" title="b3">e1</font>, <font color="Olive" title="b4">e2</font>, <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b6">v2</font>,<font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b4">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b6">v2</font>,<font color="Maroon" title="c1">G2</font> &amp;  not <!--HUMANRECFAILED-->( <font color="Olive" title="b3">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp; <font color="Olive" title="b4">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> ) holds <br/>(  not <font color="Olive" title="b3">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> &amp;  not <font color="Olive" title="b4">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">E</font> ) ) holds <br/><font color="Olive" title="b2">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G1</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">E</font>  st <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c1">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  &amp; (  for <font color="Olive" title="b2">e1</font>, <font color="Olive" title="b3">e2</font>, <font color="Olive" title="b4">v1</font>, <font color="Olive" title="b5">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b5">v2</font>,<font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b5">v2</font>,<font color="Maroon" title="c1">G2</font> &amp;  not <!--HUMANRECFAILED-->( <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Olive" title="b3">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> ) holds <br/>(  not <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> &amp;  not <font color="Olive" title="b3">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> ) ) holds <br/><font color="Olive" title="b1">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">G1</font> be    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">E</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c1">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  &amp; (  for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font>, <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c1">G2</font> &amp;  not <!--HUMANRECFAILED-->( <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Olive" title="b2">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> ) holds <br/>(  not <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> &amp;  not <font color="Olive" title="b2">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> ) ) implies <font color="Maroon" title="c3">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font>
 <span class="kw">and </span><br/><a NAME="E2:33"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c1">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> 
 <span class="kw">and </span><br/><a NAME="E3:33"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
 for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font>, <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c1">G2</font> &amp; <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c1">G2</font> &amp;  not <!--HUMANRECFAILED-->( <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Olive" title="b2">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> ) holds <br/>(  not <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> &amp;  not <font color="Olive" title="b2">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> </span><br/>

<a NAME="E4:33"/>
 for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font>, <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c3">G1</font> &amp; <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c3">G1</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">e1</font>, <font color="Maroon" title="c5">e2</font>, <font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">v2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c3">G1</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c3">G1</font> implies <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
( <font color="Maroon" title="c4">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c3">G1</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c3">G1</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font></span><br/>

<a NAME="E2:33_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c3">G1</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c3">G1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/>
<a NAME="E3:33_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T9" target="_self" onmouseover="rs('glib_007/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:33_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> ) or (  not <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> &amp;  not <font color="Maroon" title="c5">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E30">A3</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_1"/>
( <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font></span><br/><div class="add"><a NAME="E2:33_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c1">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="glib_007.html#T7" target="_self" onmouseover="rs('glib_007/T7')" onmouseout="rh()">Th7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:33_1_1_1"/>
<font color="Maroon" title="c4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="glib_000.html#D21" onmouseover="rs('glib_000/D21')" onmouseout="rh()">GLIB_000:def 21</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="33_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:33_1_1_2"/>
(  not <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> &amp;  not <font color="Maroon" title="c5">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font></span><br/><div class="add"><a NAME="E2:33_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="glib_007.html#T8" target="_self" onmouseover="rs('glib_007/T8')" onmouseout="rh()">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:33_1_1_2"/>
<font color="Maroon" title="c4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="glib_000.html#D21" onmouseover="rs('glib_000/D21')" onmouseout="rh()">GLIB_000:def 21</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:33"/>
<font color="Maroon" title="c3">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D21" onmouseover="rs('glib_000/D21')" onmouseout="rh()">GLIB_000:def 21</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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