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<span class="kw">let </span><font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b3">e</font>, <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">w</font> holds <br/>( <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">w</font>,<font color="Olive" title="b2">G1</font> iff <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">w</font>,<font color="Maroon" title="c1">G2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">G1</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">w</font> holds <br/>( <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>,<font color="Olive" title="b1">G1</font> iff <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>,<font color="Maroon" title="c1">G2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">G1</font> be    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">w</font> holds <br/>( <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>,<font color="Maroon" title="c3">G1</font> iff <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>,<font color="Maroon" title="c1">G2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">e</font>, <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">w</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="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">w</font> implies ( <font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c3">G1</font> iff <font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">G2</font> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">w</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="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c3">G1</font> iff <font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">G2</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"> ( <font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">G2</font> implies <font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c3">G1</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:42_1"/>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c3">G1</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="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">G2</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c3">G1</font> or <font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c3">G1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1"/>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c3">G1</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="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">G2</font></span><br/><div class="add"><a NAME="E2:42_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="glib_012.html#T16" target="_self" onmouseover="rs('glib_012/T16')" onmouseout="rh()">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_1_1_1"/>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_2"/>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c3">G1</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="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">G2</font></span><br/><div class="add"><a NAME="E2:42_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="glib_012.html#T16" target="_self" onmouseover="rs('glib_012/T16')" onmouseout="rh()">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_1_1_2"/>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</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></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:42"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">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="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c3">G1</font></span><br/>

<a NAME="E4:42"/>
( <font color="Maroon" title="c5">v</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  &amp; <font color="Maroon" title="c6">w</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:42"/>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="glib_006.html#T70" onmouseover="rs('glib_006/T70')" onmouseout="rh()">GLIB_006:70</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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