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

<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <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="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> implies ( ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:184"/><span class="lab"><font color="Green" title="E73">A1</font></span>: 
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</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="c1">G1</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) )</span><br/>

<a NAME="E2:184"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A2</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D34" target="_self" title="GLIB_000:def.34">Def36</a></span>;<br/>
<a NAME="E3:184"/><span class="lab"><font color="Green" title="E75">A3</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:184"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="glib_000.html#D34" target="_self" title="GLIB_000:def.34">Def36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:184"/>
( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:184"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="ref" href="glib_000.html#D17" target="_self" title="GLIB_000:def.17">Def19</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="c1">G1</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) )</span><br/>

<a NAME="E5:184"/><span class="lab"><font color="Green" title="E76">A4</font></span>: 
(  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G2</font> &amp;  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:184"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="glib_000.html#D34" target="_self" title="GLIB_000:def.34">Def36</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E77">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="184_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:184_1"/>
<font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> 
 ; <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="c2">G2</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> </span><br/><a NAME="E2:184_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> or  not <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:184"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E5:184"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="ref" href="glib_000.html#D18" target="_self" title="GLIB_000:def.18">Def20</a></span>;<br/><span class="kw">hence </span><a NAME="E3:184_1"/>
<font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D18" target="_self" title="GLIB_000:def.18">Def20</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="E7:184"/>
( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  )
 ; <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="c1">G1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) )</span><br/>

<div><span class="kw">hereby </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="c1">G1</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:184_2"/>
<font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> 
 ; <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="c2">G2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> </span><br/><a NAME="E2:184_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:184"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="ref" href="glib_000.html#D19" target="_self" title="GLIB_000:def.19">Def21</a></span>;<br/><span class="kw">hence </span><a NAME="E3:184_2"/>
<font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D19" target="_self" title="GLIB_000:def.19">Def21</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>

<div><span class="lab"><font color="Green" title="E78">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="184_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:184_3"/><span class="lab"><font color="Green" title="E79">A7</font></span>: 
<font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> 
 ; <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="c2">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="184_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">e1</font>, <font color="Maroon" title="c4">e2</font>, <font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">v2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c4">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">G2</font> implies <font color="Maroon" title="c3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:184_3_1"/>
( <font color="Maroon" title="c3">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c4">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">G2</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">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e2</font></span><br/><a NAME="E2:184_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c4">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c1">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:184"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="glib_000.html#T88" target="_self" title="GLIB_000:th.88">Th91</a></span>;<br/><span class="kw">hence </span><a NAME="E3:184_3_1"/>
<font color="Maroon" title="c3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:184_3"><span class="lab"><font color="Green" title="E79">A7</font></span></a>, <a class="ref" href="glib_000.html#D20" target="_self" title="GLIB_000:def.20">Def22</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="E3:184_3"/>
<font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D20" target="_self" title="GLIB_000:def.20">Def22</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="E10:184"/>
( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  )
 ; <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="c1">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) )</span><br/>

<div><span class="lab"><font color="Green" title="E79">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="184_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:184_4"/><span class="lab"><font color="Green" title="E80">A9</font></span>: 
<font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> 
 ; <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="c2">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="184_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">e1</font>, <font color="Maroon" title="c4">e2</font>, <font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">v2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c4">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">G2</font> implies <font color="Maroon" title="c3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:184_4_1"/>
( <font color="Maroon" title="c3">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c4">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">G2</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">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e2</font></span><br/><a NAME="E2:184_4_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c4">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c1">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:184"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="glib_000.html#T88" target="_self" title="GLIB_000:th.88">Th91</a></span>;<br/><span class="kw">hence </span><a NAME="E3:184_4_1"/>
<font color="Maroon" title="c3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:184_4"><span class="lab"><font color="Green" title="E80">A9</font></span></a>, <a class="ref" href="glib_000.html#D21" target="_self" title="GLIB_000:def.21">Def23</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="E3:184_4"/>
<font color="Maroon" title="c2">G2</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" target="_self" title="GLIB_000:def.21">Def23</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="E12:184"/>
( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  )
 ; <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="c1">G1</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E13:184"/>
( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:184"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="txt" href="#E9:184"><span class="lab"><font color="Green" title="E78">A6</font></span></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="c1">G1</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E14:184"/>
( <font color="Maroon" title="c1">G1</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c2">G2</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:184"><span class="lab"><font color="Green" title="E77">A5</font></span></a>, <a class="txt" href="#E11:184"><span class="lab"><font color="Green" title="E79">A8</font></span></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>
