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<span class="kw">let </span><font color="Maroon" title="c1">G1</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" 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_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b1">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b1">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b1">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Olive" title="b2">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G2</font> be    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" 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">set</a>  holds <br/> ( <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Olive" title="b1">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" 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="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> )</span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="171_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:171_1"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font></span><br/><a NAME="E2:171_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A2</font></span>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D32" target="_self" title="GLIB_000:def.32">Def34</a></span>;<br/><a NAME="E3:171_1"/>
( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:171_1"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="glib_000.html#D26" target="_self" title="GLIB_000:def.26">Def28</a></span>;<br/><span class="kw">hence </span><a NAME="E4:171_1"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:171_1"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="glib_000.html#D26" target="_self" title="GLIB_000:def.26">Def28</a></span>; <a class="txt" 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="E2:171"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
<font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" 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="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> )</span><br/>

<a NAME="E3:171"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.edgesInto</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="171_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:171_2"/><span class="lab"><font color="Green" title="E69">A5</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font></span><br/><a NAME="E2:171_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A6</font></span>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D32" target="_self" title="GLIB_000:def.32">Def34</a></span>;<br/><a NAME="E3:171_2"/>
( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:171_2"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="glib_000.html#D27" target="_self" title="GLIB_000:def.27">Def29</a></span>;<br/><span class="kw">hence </span><a NAME="E4:171_2"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:171_2"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="ref" href="glib_000.html#D27" target="_self" title="GLIB_000:def.27">Def29</a></span>; <a class="txt" 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="E5:171"/><span class="lab"><font color="Green" title="E69">A7</font></span>: 
<font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" 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="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> )</span><br/>

<a NAME="E6:171"/><span class="kw">then </span>
<font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesOutOf</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:171"/>
<font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:171"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>; <a class="txt" 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="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font></span><br/>

<span class="kw">thus </span><a NAME="E8:171"/>
<font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:171"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E5:171"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>; <a class="txt" 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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