<?xml version="1.0"?>
<div class="add">

<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="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> iff <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">V1</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">V2</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font>;<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">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  implies <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:71_1"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <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="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a> </span><br/><a NAME="E2:71_1"/>
( <font color="Maroon" title="c1">G1</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</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="c1">G1</font></span>)</span> &amp; <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="c2">G2</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="c2">G2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T100" onmouseover="rs('glib_000/T100')" onmouseout="rh()">GLIB_000:100</a></span>;<br/><a NAME="E3:71_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A2</font></span>: 
( <font color="Maroon" title="c1">G1</font> is   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c2">G2</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="c2">G2</font></span>)</span> &amp; <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>,<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="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="glib_000.html#T101" onmouseover="rs('glib_000/T101')" onmouseout="rh()">GLIB_000:101</a></span>;<br/><a NAME="E4:71_1"/>
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</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">G1</font> &amp;  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</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="c2">G2</font> )
 ;<br/><a NAME="E5:71_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">G1</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="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  &amp; <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="txt" href="#E3:71_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="glenum00.html#T48" target="_self" onmouseover="rs('glenum00/T48')" onmouseout="rh()">Th48</a></span>;<br/><span class="kw">hence </span><a NAME="E6:71_1"/>
<font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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">assume </span><a NAME="E2:71"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font></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="E4:71"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c1">G1</font> is   <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#T48" target="_self" onmouseover="rs('glenum00/T48')" onmouseout="rh()">Th48</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">V1</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="E6:71"/><span class="lab"><font color="Green" title="E47">A5</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="c6">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="glenum00.html#T48" target="_self" onmouseover="rs('glenum00/T48')" onmouseout="rh()">Th48</a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:71"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:71"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E6:71"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="glib_000.html#T87" onmouseover="rs('glib_000/T87')" onmouseout="rh()">GLIB_000:87</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>
