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

<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="Red">::  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>, <font color="Olive" title="b3">Y</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#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</font> &amp; <font color="Olive" title="b1">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</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#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">Y</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="173_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:173_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><br/><a NAME="E2:173_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D32" target="_self" title="GLIB_000:def.32">Def32</a></span>;<br/><a NAME="E3:173_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T75" target="_self" title="GLIB_000:th.75">Th75</a></span>;<br/><span class="kw">hence </span><a NAME="E4:173_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D32" target="_self" title="GLIB_000:def.32">Def32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:173"/>
<font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="173_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:173_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font></span><br/><a NAME="E2:173_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" target="_self" title="GLIB_000:def.33">Def33</a></span>;<br/><a NAME="E3:173_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T75" target="_self" title="GLIB_000:th.75">Th75</a></span>;<br/><span class="kw">hence </span><a NAME="E4:173_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" target="_self" title="GLIB_000:def.33">Def33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:173"/>
<font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K23" title="GLIB_000:func.23">.edgesDBetween</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
