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<span class="kw">let </span><font color="Maroon" title="c1">G1</font> be   non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  <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">  for <font color="Olive" title="b1">G2</font> being    <a href="glib_012.html#M4" title="GLIB_012:mode.4">DGraphComplement</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b3">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G2</font>  st <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b2">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  holds <br/> not <font color="Olive" title="b3">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">G2</font> be    <a href="glib_012.html#M4" title="GLIB_012:mode.4">DGraphComplement</a> of <font color="Maroon" title="c1">G1</font>; <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">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b1">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  holds <br/> not <font color="Olive" title="b2">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">v1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font>; <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">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v2</font> &amp; <font color="Maroon" title="c3">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  holds <br/> not <font color="Olive" title="b1">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>; <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="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> &amp; <font color="Maroon" title="c3">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies  not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:153"/><span class="lab"><font color="Green" title="E71">A1</font></span>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font>
 ; <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"> (  not <font color="Maroon" title="c3">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  or  not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:153"/><span class="lab"><font color="Green" title="E72">A2</font></span>: 
<font color="Maroon" title="c3">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</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">  not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">v9</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E3:153"/>
<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">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T20" onmouseover="rs('glib_000/T20')" onmouseout="rh()">GLIB_000:20</a></span>;<br/>
<a NAME="E4:153"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A3</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">v9</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> ;<br/>
<a NAME="E6:153"/><span class="lab"><font color="Green" title="E74">A4</font></span>: 
<font color="Maroon" title="c6">v9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:153"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T56" onmouseover="rs('zfmisc_1/T56')" onmouseout="rh()">ZFMISC_1:56</a></span>;<br/>
<a NAME="E7:153"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c6">v9</font>,<font color="Maroon" title="c1">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:153"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="ref" href="glib_000.html#T144" onmouseover="rs('glib_000/T144')" onmouseout="rh()">GLIB_000:144</a></span>;<br/>
<a NAME="E8:153"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c6">v9</font>,<font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:153"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="ref" href="glib_012.html#T80" target="_self" onmouseover="rs('glib_012/T80')" onmouseout="rh()">Th80</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:153"/>
 not <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:153"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="ref" href="glib_000.html#T144" onmouseover="rs('glib_000/T144')" onmouseout="rh()">GLIB_000:144</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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