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<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">G</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">  not <font color="Maroon" title="c3">G1</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> , <font color="Maroon" title="c5">f</font> being   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:64_2_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
( <font color="Maroon" title="c4">E</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">E</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> &amp;  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">G1</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">G</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c3">G1</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">G</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D5" target="_self" onmouseover="rs('glib_012/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_2_1_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">  ex <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c3">G1</font></span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c6">v</font> =  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">V</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c7">v</font> = <font color="Maroon" title="c6">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">  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c3">G1</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c8">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:64_2_1_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
( <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c9">e</font> = <font color="Maroon" title="c8">e</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="c9">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c3">G1</font></span><br/><a NAME="E4:64_2_1_1"/><span class="lab"><font color="Green" title="E32">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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E3:64_2_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>;<br/><a NAME="E5:64_2_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E3:64_2_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:64_2_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_2_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E4:64_2_1_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="glib_000.html#D14" onmouseover="rs('glib_000/D14')" onmouseout="rh()">GLIB_000:def 14</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="E4:64_2_1"/>
 not <font color="Maroon" title="c3">G1</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T136" onmouseover="rs('glib_000/T136')" onmouseout="rh()">GLIB_000:136</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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