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<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">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"> ( <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  iff  ex <font color="Olive" title="b1">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">G1</font></span>)</span> st <font color="Maroon" title="c2">G2</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b1">V</font> )</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"> (  ex <font color="Olive" title="b1">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">G1</font></span>)</span> st <font color="Maroon" title="c2">G2</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b1">V</font> implies <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:70_1"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</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">  ex <font color="Olive" title="b1">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">G1</font></span>)</span> st <font color="Maroon" title="c2">G2</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b1">V</font></span><br/><a NAME="E2:70_1"/>
<font color="Maroon" title="c2">G2</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="c2">G2</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/><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">G1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:70_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c2">G2</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_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">G1</font>,<font color="Maroon" title="c3">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="glenum00.html#T45" target="_self" onmouseover="rs('glenum00/T45')" onmouseout="rh()">Th45</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c4">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="c2">G2</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c4">V</font></span><br/><a NAME="E5:70_1"/>
<font color="Maroon" title="c2">G2</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</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="E6:70_1"/>
<font color="Maroon" title="c2">G2</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c4">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="glib_000.html#T101" onmouseover="rs('glib_000/T101')" onmouseout="rh()">GLIB_000:101</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">given </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">G1</font></span>)</span><span class="kw"> such that </span><a NAME="E3:70"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<font color="Maroon" title="c2">G2</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</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="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_2"><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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:70_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">V2</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="c2">G2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:70_2"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c4">x</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="c2">G2</font>,<font color="Maroon" title="c5">V2</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/><a NAME="E4:70_2"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E44">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>;<br/><a NAME="E5:70_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A5</font></span>: 
<font color="Maroon" title="c5">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font>
 ;<br/><a NAME="E6:70_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A6</font></span>: 
<font color="Maroon" title="c5">V2</font> is   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">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E7:70_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c5">V2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="#E3:70_2"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E5:70_2"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="chord.html#T29" onmouseover="rs('chord/T29')" onmouseout="rh()">CHORD:29</a></span>;<br/><span class="kw">hence </span><a NAME="E8:70_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_2"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E6:70_2"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="glenum00.html#T45" target="_self" onmouseover="rs('glenum00/T45')" onmouseout="rh()">Th45</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="E5:70"/>
<font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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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