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<span class="kw">let </span><font color="Maroon" title="c1">G2</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">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">G1</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">V</font>  st <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> holds <br/><font color="Maroon" title="c1">G2</font> is   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Olive" title="b3">G1</font>,<font color="Olive" title="b1">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">V</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">G1</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</font>  st <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> holds <br/><font color="Maroon" title="c1">G2</font> is   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Olive" title="b1">G1</font>,<font color="Maroon" title="c2">v</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">G1</font> be    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</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">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> implies <font color="Maroon" title="c1">G2</font> is   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">v</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:85"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
( <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">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="c1">G2</font> is   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">v</font></span><br/>

<a NAME="E2:85"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c1">G2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T57" onmouseover="rs('glib_006/T57')" onmouseout="rh()">GLIB_006:57</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">V1</font> = <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="c4">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E3:85"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c4">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" target="_self" onmouseover="rs('glib_007/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:85"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A3</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T117" onmouseover="rs('zfmisc_1/T117')" onmouseout="rh()">ZFMISC_1:117</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">V1</font> = <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="c4">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   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="c4">G1</font></span>)</span> ;<br/>
<a NAME="E6:85"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c6">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E4:85"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="glib_007.html#T53" target="_self" onmouseover="rs('glib_007/T53')" onmouseout="rh()">Th53</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:85"/>
<font color="Maroon" title="c1">G2</font> is   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c4">G1</font>,<font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:85"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E4:85"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</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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