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<b>let </b><font color="Maroon">G1</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">G2</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/>



<b>assume </b><a NAME="E1:182"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">G1</font> <a href="glib_000.html#R5" target="_self">==</a> <font color="Maroon">G2</font>
 ;<br/>

<a NAME="E2:182"/><b>then </b><i><font color="Green">E29</font></i>: 
(  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G2</font> &amp;  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G2</font> &amp;  <a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G1</font> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G2</font> &amp;  <a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G1</font> <a href="funct_2.html#R1">=</a>  <a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G2</font> )
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:182_1"/>
<font color="Maroon">G1</font> is <a href="glib_000.html#V2" target="_self">finite</a>
 ;<br/><a NAME="E2:182_1"/><b>then </b>
(  <a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G2</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G2</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:182_1"/>
<font color="Maroon">G2</font> is <a href="glib_000.html#V2" target="_self">finite</a>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="182_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:182_2"/>
<font color="Maroon">G1</font> is <a href="glib_000.html#V3" target="_self">loopless</a>
 ;<br/><a NAME="E2:182_2"/><b>then </b>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7" target="_self">the_Edges_of</a> <font color="Maroon">G2</font> or  not <span class="p1">(<span class="default"><a href="glib_000.html#K10" target="_self">the_Source_of</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" target="_self">the_Target_of</a> <font color="Maroon">G2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:182_2"/>
<font color="Maroon">G2</font> is <a href="glib_000.html#V3" target="_self">loopless</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:182"/>
( <font color="Maroon">G1</font> is <a href="glib_000.html#V3" target="_self">loopless</a> implies <font color="Maroon">G2</font> is <a href="glib_000.html#V3" target="_self">loopless</a> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:182_3"/>
<font color="Maroon">G1</font> is <a href="glib_000.html#V4" target="_self">trivial</a>
 ;<br/><a NAME="E2:182_3"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" target="_self">the_Vertices_of</a> <font color="Maroon">G2</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:182_3"/>
<font color="Maroon">G2</font> is <a href="glib_000.html#V4" target="_self">trivial</a>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="182_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:182_4"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">G1</font> is <a href="glib_000.html#V5" target="_self">non-multi</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="182_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">e2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:182_4_1"/>
( <font color="Maroon">e1</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G2</font> &amp; <font color="Maroon">e2</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G2</font> )
 ;<br/><a NAME="E2:182_4_1"/><b>then </b>
( <font color="Maroon">e1</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G1</font> &amp; <font color="Maroon">e2</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G1</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:182_4_1"/>
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:182_4"/>
<font color="Maroon">G2</font> is <a href="glib_000.html#V5" target="_self">non-multi</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:182"/>
( <font color="Maroon">G1</font> is <a href="glib_000.html#V5" target="_self">non-multi</a> implies <font color="Maroon">G2</font> is <a href="glib_000.html#V5" target="_self">non-multi</a> )
 ;<br/>

<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="182_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:182_5"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">G1</font> is <a href="glib_000.html#V6" target="_self">non-Dmulti</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="182_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">e2</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:182_5_1"/>
( <font color="Maroon">e1</font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G2</font> &amp; <font color="Maroon">e2</font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G2</font> )
 ;<br/><a NAME="E2:182_5_1"/><b>then </b>
( <font color="Maroon">e1</font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G1</font> &amp; <font color="Maroon">e2</font> <a href="glib_000.html#R2" target="_self">DJoins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G1</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:182_5_1"/>
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D34" target="_self">Def34</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:182_5"/>
<font color="Maroon">G2</font> is <a href="glib_000.html#V6" target="_self">non-Dmulti</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:182"/>
( <font color="Maroon">G1</font> is <a href="glib_000.html#V6" target="_self">non-Dmulti</a> implies <font color="Maroon">G2</font> is <a href="glib_000.html#V6" target="_self">non-Dmulti</a> )
 ;<br/>

<b>thus </b><a NAME="E11:182"/>
( <font color="Maroon">G1</font> is <a href="glib_000.html#V7" target="_self">simple</a> implies <font color="Maroon">G2</font> is <a href="glib_000.html#V7" target="_self">simple</a> )
 <b>by </b><i>, , </i>;<br/>

<b>thus </b><a NAME="E12:182"/>
( <font color="Maroon">G1</font> is <a href="glib_000.html#V8" target="_self">Dsimple</a> implies <font color="Maroon">G2</font> is <a href="glib_000.html#V8" target="_self">Dsimple</a> )
 <b>by </b><i>, , </i>;<br/>


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