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<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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:78"/><span class="lab"><font color="Green" title="E50">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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:78_1"/><span class="lab"><font color="Green" title="E51">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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V3" title="GLIB_014:attr.3">loopless</a> </span><br/>

<a NAME="E2:78_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#K7" title="GLENUM00:func.7">.allInducedSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T55" target="_self" onmouseover="rs('glenum00/T55')" onmouseout="rh()">Th55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:78_1"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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="#E1:78_1"><span class="lab"><font color="Green" title="E51">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="E2:78"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:78_2"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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="E2:78_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#K7" title="GLENUM00:func.7">.allInducedSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T55" target="_self" onmouseover="rs('glenum00/T55')" onmouseout="rh()">Th55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:78_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="#E1:78_2"><span class="lab"><font color="Green" title="E52">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="E3:78"/><span class="lab"><font color="Green" title="E52">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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:78_3"/>
<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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V4" title="GLIB_014:attr.4">non-multi</a> </span><br/>

<a NAME="E2:78_3"/><span class="kw">then </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> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:78_3"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:78"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:78_4"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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="E2:78_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#K7" title="GLENUM00:func.7">.allInducedSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T47" target="_self" onmouseover="rs('glenum00/T47')" onmouseout="rh()">Th47</a></span>;<br/>
<a NAME="E3:78_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A8</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="#E1:78_4"><span class="lab"><font color="Green" title="E54">A7</font></span></a></span>;<br/>
<a NAME="E4:78_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="E5:78_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="#E3:78_4"><span class="lab"><font color="Green" title="E55">A8</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="E5:78"/><span class="lab"><font color="Green" title="E54">A9</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:78_5"/>
<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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V5" title="GLIB_014:attr.5">non-Dmulti</a> </span><br/>

<a NAME="E2:78_5"/><span class="kw">then </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> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:78_5"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:78"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:78_6"/><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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="E2:78_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#K7" title="GLENUM00:func.7">.allInducedSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T47" target="_self" onmouseover="rs('glenum00/T47')" onmouseout="rh()">Th47</a></span>;<br/>
<a NAME="E3:78_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A12</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="#E1:78_6"><span class="lab"><font color="Green" title="E56">A11</font></span></a></span>;<br/>
<a NAME="E4:78_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="E5:78_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="#E3:78_6"><span class="lab"><font color="Green" title="E57">A12</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="E7:78"/>
( <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#K7" title="GLENUM00:func.7">.allInducedSG()</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="#E1:78"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E52">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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:78"/>
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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="#E2:78"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E53">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"> ( ( <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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E9:78"/>
( <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#K7" title="GLENUM00:func.7">.allInducedSG()</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="#E1:78"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E5:78"><span class="lab"><font color="Green" title="E54">A9</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E10:78"/>
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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="#E2:78"><span class="lab"><font color="Green" title="E51">A3</font></span></a>, <a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E55">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> 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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:78_7"/>
<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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V8" title="GLIB_014:attr.8">acyclic</a> </span><br/><a NAME="E2:78_7"/><span class="kw">then </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> 
 ;<br/><span class="kw">hence </span><a NAME="E3:78_7"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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"> 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>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:78_8"/><span class="lab"><font color="Green" title="E56">A13</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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="E2:78_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#K7" title="GLENUM00:func.7">.allInducedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T47" target="_self" onmouseover="rs('glenum00/T47')" onmouseout="rh()">Th47</a></span>;<br/><a NAME="E3:78_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">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_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_8"><span class="lab"><font color="Green" title="E56">A13</font></span></a></span>;<br/><a NAME="E4:78_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="E5:78_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="#E3:78_8"><span class="lab"><font color="Green" title="E57">A14</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#K7" title="GLENUM00:func.7">.allInducedSG()</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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:78_9"/>
<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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a> </span><br/><a NAME="E2:78_9"/><span class="kw">then </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> 
 ;<br/><span class="kw">hence </span><a NAME="E3:78_9"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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"> 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="chord.html#V6" title="CHORD:attr.6">chordal</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:78_10"/><span class="lab"><font color="Green" title="E56">A15</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</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="E2:78_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#K7" title="GLENUM00:func.7">.allInducedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T47" target="_self" onmouseover="rs('glenum00/T47')" onmouseout="rh()">Th47</a></span>;<br/><a NAME="E3:78_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A16</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="#E1:78_10"><span class="lab"><font color="Green" title="E56">A15</font></span></a></span>;<br/><a NAME="E4:78_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="E5:78_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="#E3:78_10"><span class="lab"><font color="Green" title="E57">A16</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><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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:78_11"/><span class="lab"><font color="Green" title="E56">A17</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_11_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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  holds <br/><font color="Olive" title="b1">H</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  implies <font color="Maroon" title="c2">H</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:78_11_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#K7" title="GLENUM00:func.7">.allInducedSG()</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="chord.html#V6" title="CHORD:attr.6">chordal</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">V</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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="c1">G</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:78_11_1"/><span class="lab"><font color="Green" title="E57">A18</font></span>: 
<font color="Maroon" title="c2">H</font> is   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T45" target="_self" onmouseover="rs('glenum00/T45')" onmouseout="rh()">Th45</a></span>;<br/><span class="kw">thus </span><a NAME="E4:78_11_1"/>
<font color="Maroon" title="c2">H</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_11"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="txt" href="#E3:78_11_1"><span class="lab"><font color="Green" title="E57">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="E3:78_11"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D11" onmouseover="rs('glib_014/D11')" onmouseout="rh()">GLIB_014:def 11</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_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  iff <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:78_12"/><span class="lab"><font color="Green" title="E56">A19</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V11" title="GLIB_014:attr.11">chordal</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="chord.html#V6" title="CHORD:attr.6">chordal</a> </span><br/><a NAME="E2:78_12"/>
<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#K7" title="GLENUM00:func.7">.allInducedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T47" target="_self" onmouseover="rs('glenum00/T47')" onmouseout="rh()">Th47</a></span>;<br/><a NAME="E3:78_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">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="chord.html#V6" title="CHORD:attr.6">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_12"><span class="lab"><font color="Green" title="E56">A19</font></span></a></span>;<br/><a NAME="E4:78_12"/>
<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="E5:78_12"/>
<font color="Maroon" title="c1">G</font> is  <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_12"><span class="lab"><font color="Green" title="E57">A20</font></span></a>, <a class="ref" href="chord.html#T95" onmouseover="rs('chord/T95')" onmouseout="rh()">CHORD:95</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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:78_13"/><span class="lab"><font color="Green" title="E56">A21</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_13_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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  holds <br/><font color="Olive" title="b1">H</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  implies <font color="Maroon" title="c2">H</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:78_13_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#K7" title="GLENUM00:func.7">.allInducedSG()</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_012.html#V1" title="GLIB_012:attr.1">loopfull</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">V</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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="c1">G</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:78_13_1"/><span class="lab"><font color="Green" title="E57">A22</font></span>: 
<font color="Maroon" title="c2">H</font> is   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T45" target="_self" onmouseover="rs('glenum00/T45')" onmouseout="rh()">Th45</a></span>;<br/><span class="kw">thus </span><a NAME="E4:78_13_1"/>
<font color="Maroon" title="c2">H</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_13"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E3:78_13_1"><span class="lab"><font color="Green" title="E57">A22</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="E3:78_13"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D13" onmouseover="rs('glib_014/D13')" onmouseout="rh()">GLIB_014:def 13</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="E1:78_14"/><span class="lab"><font color="Green" title="E56">A23</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  is  <a href="glib_014.html#V13" title="GLIB_014:attr.13">loopfull</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_012.html#V1" title="GLIB_012:attr.1">loopfull</a> </span><br/><a NAME="E2:78_14"/>
<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#K7" title="GLENUM00:func.7">.allInducedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T47" target="_self" onmouseover="rs('glenum00/T47')" onmouseout="rh()">Th47</a></span>;<br/><a NAME="E3:78_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A24</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_012.html#V1" title="GLIB_012:attr.1">loopfull</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_14"><span class="lab"><font color="Green" title="E56">A23</font></span></a></span>;<br/><a NAME="E4:78_14"/>
<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="E5:78_14"/>
<font color="Maroon" title="c1">G</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_14"><span class="lab"><font color="Green" title="E57">A24</font></span></a>, <a class="ref" href="glib_012.html#T4" onmouseover="rs('glib_012/T4')" onmouseout="rh()">GLIB_012:4</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>


</div>
