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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   non  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  <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">e</font> being   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c1">G</font> holds <br/> (  <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a> <font color="Olive" title="b1">e</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  iff  not <font color="Olive" title="b1">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">e</font> be   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</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#K4" title="GLENUM00:func.4">createGraph</a> <font color="Maroon" title="c2">e</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  iff  not <font color="Maroon" title="c2">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  )</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"> (  not <font color="Maroon" title="c2">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  implies  <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a> <font color="Maroon" title="c2">e</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a> <font color="Maroon" title="c2">e</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</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">  not <font color="Maroon" title="c2">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> </span><br/><span class="kw">assume </span><a NAME="E2:22_1"/>
<font color="Maroon" title="c2">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:22_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_009.html#D2" onmouseover="rs('glib_009/D2')" onmouseout="rh()">GLIB_009:def 2</a></span>;<br/><a NAME="E5:22_1"/>
( <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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</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="c1">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E6:22_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="glenum00.html#T14" target="_self" onmouseover="rs('glenum00/T14')" onmouseout="rh()">Th14</a>, <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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:22"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 not <font color="Maroon" title="c2">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</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">  <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a> <font color="Maroon" title="c2">e</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> </span><br/>

<span class="kw">assume </span><a NAME="E3:22"/>
 not  <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a> <font color="Maroon" title="c2">e</font> is  <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">e0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:22"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c4">e0</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font>, <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a> <font color="Maroon" title="c2">e</font>
 <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>;<br/>
<a NAME="E6:22"/>
( <font color="Maroon" title="c4">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Maroon" title="c3">v</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="glenum00.html#T15" target="_self" onmouseover="rs('glenum00/T15')" onmouseout="rh()">Th15</a>, <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="E7:22"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="glib_000.html#T72" onmouseover="rs('glib_000/T72')" onmouseout="rh()">GLIB_000:72</a>, <a class="ref" href="glib_009.html#T45" onmouseover="rs('glib_009/T45')" onmouseout="rh()">GLIB_009:45</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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