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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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:106"/>
( <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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:106"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="106_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:106_1"/><span class="lab"><font color="Green" title="E62">A2</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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="E2:106_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#K8" title="GLENUM00:func.8">.allSpanningSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T72" target="_self" onmouseover="rs('glenum00/T72')" onmouseout="rh()">Th72</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:106_1"/>
<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="#E1:106_1"><span class="lab"><font color="Green" title="E62">A2</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="E3:106"/>
( <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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:106"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="106_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:106_2"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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="E2:106_2"/>
<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#K8" title="GLENUM00:func.8">.allSpanningSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T62" target="_self" onmouseover="rs('glenum00/T62')" onmouseout="rh()">Th62</a></span>;<br/>
<a NAME="E3:106_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A5</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="#E1:106_2"><span class="lab"><font color="Green" title="E63">A4</font></span></a></span>;<br/>
<a NAME="E4:106_2"/>
<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="E5:106_2"/>
<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="#E3:106_2"><span class="lab"><font color="Green" title="E64">A5</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="E5:106"/>
( <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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:106"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="106_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:106_3"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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="E2:106_3"/>
<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#K8" title="GLENUM00:func.8">.allSpanningSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T62" target="_self" onmouseover="rs('glenum00/T62')" onmouseout="rh()">Th62</a></span>;<br/>
<a NAME="E3:106_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">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#V6" title="GLIB_000:attr.6">non-Dmulti</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_3"><span class="lab"><font color="Green" title="E64">A7</font></span></a></span>;<br/>
<a NAME="E4:106_3"/>
<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="E5:106_3"/>
<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="#E3:106_3"><span class="lab"><font color="Green" title="E65">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="E7:106"/>
( <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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V6" title="GLIB_014:attr.6">simple</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:106"/>
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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="#E2:106"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E4:106"><span class="lab"><font color="Green" title="E62">A3</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E9:106"/>
( <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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V7" title="GLIB_014:attr.7">Dsimple</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E10:106"/>
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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="#E2:106"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="#E6:106"><span class="lab"><font color="Green" title="E63">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_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  implies <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E11:106"/>
( <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#K8" title="GLENUM00:func.8">.allSpanningSG()</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> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  iff <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:106_4"/><span class="lab"><font color="Green" title="E64">A9</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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="E2:106_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#K8" title="GLENUM00:func.8">.allSpanningSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T62" target="_self" onmouseover="rs('glenum00/T62')" onmouseout="rh()">Th62</a></span>;<br/><a NAME="E3:106_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A10</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="#E1:106_4"><span class="lab"><font color="Green" title="E64">A9</font></span></a></span>;<br/><a NAME="E4:106_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="E5:106_4"/>
<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="#E3:106_4"><span class="lab"><font color="Green" title="E65">A10</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>

<span class="kw">thus </span><a NAME="E13:106"/>
( <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#K8" title="GLENUM00:func.8">.allSpanningSG()</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> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:106_5"/><span class="lab"><font color="Green" title="E64">A11</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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="E2:106_5"/>
<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#K8" title="GLENUM00:func.8">.allSpanningSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T62" target="_self" onmouseover="rs('glenum00/T62')" onmouseout="rh()">Th62</a></span>;<br/><a NAME="E3:106_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">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_008.html#V1" title="GLIB_008:attr.1">edgeless</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_5"><span class="lab"><font color="Green" title="E64">A11</font></span></a></span>;<br/><a NAME="E4:106_5"/>
<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="E5:106_5"/>
<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="#E3:106_5"><span class="lab"><font color="Green" title="E65">A12</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>


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