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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_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">G9</font> =  the    <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#K11" title="GLENUM00:func.11">.allConnectedSG()</a> ;<br/>
<a NAME="E1:179"/>
<font color="Maroon" title="c1">G</font> is    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T35" target="_self" onmouseover="rs('glenum00/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<a NAME="E2:179"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A1</font></span>: 
 the    <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#K11" title="GLENUM00:func.11">.allConnectedSG()</a>  is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre1.html#T119" onmouseover="rs('glibpre1/T119')" onmouseout="rh()">GLIBPRE1:119</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="179_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"> <font color="Maroon" title="c1">G</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the    <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#K11" title="GLENUM00:func.11">.allConnectedSG()</a> </span></a><div class="add"><a NAME="E1:179_1"/><span class="lab"><font color="Green" title="E105">A2</font></span>: 
 for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="glenum00.html#K3" title="GLENUM00:func.3">createGraph</a> <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T127" target="_self" onmouseover="rs('glenum00/T127')" onmouseout="rh()">Th127</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:179_1_1"/>
( <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  or  not <font color="Maroon" title="c1">G</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:179_1_1_1"/>
<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> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the    <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#K11" title="GLENUM00:func.11">.allConnectedSG()</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:179_1_1_1"/>
<font color="Maroon" title="c1">G</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the    <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#K11" title="GLENUM00:func.11">.allConnectedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:179_1"><span class="lab"><font color="Green" title="E105">A2</font></span></a>, <a class="ref" href="glenum00.html#T24" target="_self" onmouseover="rs('glenum00/T24')" onmouseout="rh()">Th24</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="179_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:179_1_1_2"/>
 not <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> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the    <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#K11" title="GLENUM00:func.11">.allConnectedSG()</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">G0</font> = <font color="Maroon" title="c1">G</font> as   non  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> ;<br/><a NAME="E3:179_1_1_2"/>
 for <font color="Olive" title="b1">e</font> being   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">G0</font> holds   <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a> <font color="Olive" title="b1">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T128" target="_self" onmouseover="rs('glenum00/T128')" onmouseout="rh()">Th128</a></span>;<br/><span class="kw">hence </span><a NAME="E4:179_1_1_2"/>
<font color="Maroon" title="c1">G</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the    <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#K11" title="GLENUM00:func.11">.allConnectedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:179_1"><span class="lab"><font color="Green" title="E105">A2</font></span></a>, <a class="ref" href="glenum00.html#T25" target="_self" onmouseover="rs('glenum00/T25')" onmouseout="rh()">Th25</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">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:179"/>
<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#K11" title="GLENUM00:func.11">.allConnectedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:179"><span class="lab"><font color="Green" title="E104">A1</font></span></a>, <a class="ref" href="glib_000.html#T87" onmouseover="rs('glib_000/T87')" onmouseout="rh()">GLIB_000:87</a>, <a class="ref" href="glib_014.html#T22" onmouseover="rs('glib_014/T22')" onmouseout="rh()">GLIB_014:22</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/>


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