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

<span class="kw">let </span><font color="Maroon" title="c1">G</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"> ( ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V3" title="GLIB_014:attr.3">loopless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V3" title="GLIB_014:attr.3">loopless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:46"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V3" title="GLIB_014:attr.3">loopless</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="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V3" title="GLIB_014:attr.3">loopless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_1"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</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="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V3" title="GLIB_014:attr.3">loopless</a> </span><br/>

<a NAME="E2:46_1"/>
<font color="Maroon" title="c1">G</font> is    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T35" target="_self" onmouseover="rs('glenum00/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:46_1"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V3" title="GLIB_014:attr.3">loopless</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="glib_014.html#T23" onmouseover="rs('glib_014/T23')" onmouseout="rh()">GLIB_014:23</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>

<span class="kw">thus </span><a NAME="E2:46"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V3" title="GLIB_014:attr.3">loopless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</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="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_2"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V3" title="GLIB_014:attr.3">loopless</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="c1">G</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> </span><br/>

<a NAME="E2:46_2"/>
<font color="Maroon" title="c1">G</font> is    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T35" target="_self" onmouseover="rs('glenum00/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:46_2"/>
<font color="Maroon" title="c1">G</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="txt" href="#E1:46_2"><span class="lab"><font color="Green" title="E34">A4</font></span></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>

<span class="kw">thus </span><a NAME="E3:46"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</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="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_3"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</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="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_3_1"><span class="kw">now </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">  for <font color="Olive" title="b1">H</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  holds <br/><font color="Olive" title="b1">H</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">H</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"> ( <font color="Maroon" title="c2">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  implies <font color="Maroon" title="c2">H</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:46_3_1"/>
<font color="Maroon" title="c2">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</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="c2">H</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> </span><br/><a NAME="E2:46_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">H</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T1" target="_self" onmouseover="rs('glenum00/T1')" onmouseout="rh()">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:46_3_1"/>
<font color="Maroon" title="c2">H</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="txt" href="#E1:46_3"><span class="lab"><font color="Green" title="E35">A6</font></span></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>
<span class="kw">hence </span><a NAME="E3:46_3"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D4" onmouseover="rs('glib_014/D4')" onmouseout="rh()">GLIB_014:def 4</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>

<span class="kw">thus </span><a NAME="E4:46"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</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="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_4"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</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="c1">G</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> </span><br/>

<a NAME="E2:46_4"/>
<font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T3" target="_self" onmouseover="rs('glenum00/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E3:46_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> 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="txt" href="#E1:46_4"><span class="lab"><font color="Green" title="E36">A8</font></span></a></span>;<br/>
<a NAME="E4:46_4"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T128" onmouseover="rs('glib_000/T128')" onmouseout="rh()">GLIB_000:128</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:46_4"/>
<font color="Maroon" title="c1">G</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="txt" href="#E3:46_4"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="glib_000.html#T89" onmouseover="rs('glib_000/T89')" onmouseout="rh()">GLIB_000:89</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>

<span class="kw">thus </span><a NAME="E5:46"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</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="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_5"/><span class="lab"><font color="Green" title="E37">A11</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</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="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_5_1"><span class="kw">now </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">  for <font color="Olive" title="b1">H</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  holds <br/><font color="Olive" title="b1">H</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">H</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"> ( <font color="Maroon" title="c2">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  implies <font color="Maroon" title="c2">H</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:46_5_1"/>
<font color="Maroon" title="c2">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</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="c2">H</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> </span><br/><a NAME="E2:46_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">H</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T1" target="_self" onmouseover="rs('glenum00/T1')" onmouseout="rh()">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:46_5_1"/>
<font color="Maroon" title="c2">H</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="txt" href="#E1:46_5"><span class="lab"><font color="Green" title="E37">A11</font></span></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>
<span class="kw">hence </span><a NAME="E3:46_5"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D5" onmouseover="rs('glib_014/D5')" onmouseout="rh()">GLIB_014:def 5</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>

<span class="kw">thus </span><a NAME="E6:46"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</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="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:46_6"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</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="c1">G</font> is  <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> </span><br/>

<a NAME="E2:46_6"/>
<font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T3" target="_self" onmouseover="rs('glenum00/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E3:46_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A14</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> 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="txt" href="#E1:46_6"><span class="lab"><font color="Green" title="E38">A13</font></span></a></span>;<br/>
<a NAME="E4:46_6"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T128" onmouseover="rs('glib_000/T128')" onmouseout="rh()">GLIB_000:128</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:46_6"/>
<font color="Maroon" title="c1">G</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="txt" href="#E3:46_6"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="glib_000.html#T89" onmouseover="rs('glib_000/T89')" onmouseout="rh()">GLIB_000:89</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>

<span class="kw">thus </span><a NAME="E7:46"/>
( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E34">A5</font></span></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"> ( ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:46"/>
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</a>  implies <font color="Maroon" title="c1">G</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="#E2:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E35">A7</font></span></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"> ( ( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E9:46"/>
( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E36">A10</font></span></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"> ( ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E10:46"/>
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</a>  implies <font color="Maroon" title="c1">G</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="#E2:46"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E37">A12</font></span></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"> ( ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<div><span class="kw">hereby </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"> ( ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:46_7"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</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="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_7_1"><span class="kw">now </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">  for <font color="Olive" title="b1">H</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  holds <br/><font color="Olive" title="b1">H</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">H</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"> ( <font color="Maroon" title="c2">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  implies <font color="Maroon" title="c2">H</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:46_7_1"/>
<font color="Maroon" title="c2">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</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="c2">H</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </span><br/><a NAME="E2:46_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">H</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T1" target="_self" onmouseover="rs('glenum00/T1')" onmouseout="rh()">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:46_7_1"/>
<font color="Maroon" title="c2">H</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_7"><span class="lab"><font color="Green" title="E38">A15</font></span></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><span class="kw">hence </span><a NAME="E3:46_7"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D8" onmouseover="rs('glib_014/D8')" onmouseout="rh()">GLIB_014:def 8</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">hereby </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"> ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  iff <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:46_8"/><span class="lab"><font color="Green" title="E38">A16</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</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="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </span><br/><a NAME="E2:46_8"/>
<font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T3" target="_self" onmouseover="rs('glenum00/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E3:46_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A17</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_8"><span class="lab"><font color="Green" title="E38">A16</font></span></a></span>;<br/><a NAME="E4:46_8"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T128" onmouseover="rs('glib_000/T128')" onmouseout="rh()">GLIB_000:128</a></span>;<br/><span class="kw">hence </span><a NAME="E5:46_8"/>
<font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_8"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="glib_002.html#T44" onmouseover="rs('glib_002/T44')" onmouseout="rh()">GLIB_002:44</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">hereby </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"> ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:46_9"/><span class="lab"><font color="Green" title="E38">A18</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</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="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="46_9_1"><span class="kw">now </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">  for <font color="Olive" title="b1">H</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  holds <br/><font color="Olive" title="b1">H</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">H</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"> ( <font color="Maroon" title="c2">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  implies <font color="Maroon" title="c2">H</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:46_9_1"/>
<font color="Maroon" title="c2">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</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="c2">H</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> </span><br/><a NAME="E2:46_9_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">H</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T1" target="_self" onmouseover="rs('glenum00/T1')" onmouseout="rh()">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:46_9_1"/>
<font color="Maroon" title="c2">H</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_9"><span class="lab"><font color="Green" title="E38">A18</font></span></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><span class="kw">hence </span><a NAME="E3:46_9"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D12" onmouseover="rs('glib_014/D12')" onmouseout="rh()">GLIB_014:def 12</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">hereby </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>
<div class="add"><span class="kw">assume </span><a NAME="E1:46_10"/><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</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="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> </span><br/><a NAME="E2:46_10"/>
<font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T3" target="_self" onmouseover="rs('glenum00/T3')" onmouseout="rh()">Th3</a></span>;<br/><a NAME="E3:46_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A20</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_10"><span class="lab"><font color="Green" title="E38">A19</font></span></a></span>;<br/><a NAME="E4:46_10"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T128" onmouseover="rs('glib_000/T128')" onmouseout="rh()">GLIB_000:128</a></span>;<br/><span class="kw">hence </span><a NAME="E5:46_10"/>
<font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_10"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="glib_008.html#T52" onmouseover="rs('glib_008/T52')" onmouseout="rh()">GLIB_008:52</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>
