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

<span class="kw">let </span><font color="Maroon" title="c1">G</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">H</font> being    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> holds   <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Olive" title="b1">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font> be    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</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">  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c2">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span></span><br/>



<a NAME="E1:62"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c2">H</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T29" target="_self" onmouseover="rs('glenum00/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">R</font> = <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span> as   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">Relation</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span> ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1"><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">H1</font>, <font color="Olive" title="b2">H2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">H</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  holds <br/> ( ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c2">H</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">R</font> ) &amp; ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">R</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c2">H</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">H1</font>, <font color="Maroon" title="c5">H2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">H</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</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"> ( ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c2">H</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">R</font> ) &amp; ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">R</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c2">H</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"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">R</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c2">H</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:62_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c2">H</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"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">R</font></span><br/><a NAME="E2:62_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c4">H1</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c5">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#D6" target="_self" onmouseover="rs('glenum00/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E3:62_1_1"/>
( <font color="Maroon" title="c4">H1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  &amp; <font color="Maroon" title="c5">H2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><a NAME="E4:62_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="glenum00.html#D6" target="_self" onmouseover="rs('glenum00/D6')" onmouseout="rh()">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:62_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mmlquer2.html#T4" onmouseover="rs('mmlquer2/T4')" onmouseout="rh()">MMLQUER2:4</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="E2:62_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">R</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"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c2">H</font></span><br/><a NAME="E3:62_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mmlquer2.html#T4" onmouseover="rs('mmlquer2/T4')" onmouseout="rh()">MMLQUER2:4</a></span>;<br/><a NAME="E4:62_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">H1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c5">H2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T15" onmouseover="rs('relat_1/T15')" onmouseout="rh()">RELAT_1:15</a></span>;<br/><a NAME="E5:62_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">H1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  &amp; <font color="Maroon" title="c5">H2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T40" target="_self" onmouseover="rs('glenum00/T40')" onmouseout="rh()">Th40</a></span>;<br/><a NAME="E6:62_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">H1</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c5">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="glenum00.html#D6" target="_self" onmouseover="rs('glenum00/D6')" onmouseout="rh()">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E7:62_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">H1</font>,<font color="Maroon" title="c5">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#D6" target="_self" onmouseover="rs('glenum00/D6')" onmouseout="rh()">Def6</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="E4:62"/>
 <a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c2">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#D2" onmouseover="rs('relset_1/D2')" onmouseout="rh()">RELSET_1:def 2</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>
