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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">S</font>, <font color="Olive" title="b3">T</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">E</font>,<font color="Maroon" title="c1">V</font>  st  ex <font color="Olive" title="b4">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  or <font color="Olive" title="b3">T</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  ) holds <br/> not  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">S</font>,<font color="Olive" title="b3">T</font>) is  <a href="glib_013.html#V5" title="GLIB_013:attr.5">locally-finite</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</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">S</font>, <font color="Olive" title="b2">T</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">V</font>  st  ex <font color="Olive" title="b3">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  or <font color="Olive" title="b2">T</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  ) holds <br/> not  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">E</font>,<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font>) is  <a href="glib_013.html#V5" title="GLIB_013:attr.5">locally-finite</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font>, <font color="Maroon" title="c4">T</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">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">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c3">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  or <font color="Maroon" title="c4">T</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  ) implies  not  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>) is  <a href="glib_013.html#V5" title="GLIB_013:attr.5">locally-finite</a>  )</span><br/>





<span class="kw">given </span><font color="Maroon" title="c5">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><a NAME="E2:59"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
( <font color="Maroon" title="c3">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  or <font color="Maroon" title="c4">T</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</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  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>) is  <a href="glib_013.html#V5" title="GLIB_013:attr.5">locally-finite</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">G</font> =  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">v</font> = <font color="Maroon" title="c5">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>)</span>)</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_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="E1:59_1"/>
( <font color="Maroon" title="c3">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  or <font color="Maroon" title="c4">T</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E22">A1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><span class="kw">suppose </span></a><a NAME="E1:59_1_1"/>
<font color="Maroon" title="c3">S</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</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  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>) is  <a href="glib_013.html#V5" title="GLIB_013:attr.5">locally-finite</a> </span><br/><div class="add"><a NAME="E2:59_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>)</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> 
 ;<br/><a NAME="E3:59_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre0.html#T25" onmouseover="rs('glibpre0/T25')" onmouseout="rh()">GLIBPRE0:25</a></span>;<br/><span class="kw">hence </span><a NAME="E4:59_1_1"/>
 not  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>) is  <a href="glib_013.html#V5" title="GLIB_013:attr.5">locally-finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#T25" target="_self" onmouseover="rs('glib_013/T25')" onmouseout="rh()">Th25</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="59_1_2"><span class="kw">suppose </span></a><a NAME="E1:59_1_2"/>
<font color="Maroon" title="c4">T</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</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  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>) is  <a href="glib_013.html#V5" title="GLIB_013:attr.5">locally-finite</a> </span><br/><div class="add"><a NAME="E2:59_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>)</span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> 
 ;<br/><a NAME="E3:59_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">v</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre0.html#T25" onmouseover="rs('glibpre0/T25')" onmouseout="rh()">GLIBPRE0:25</a></span>;<br/><span class="kw">hence </span><a NAME="E4:59_1_2"/>
 not  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>) is  <a href="glib_013.html#V5" title="GLIB_013:attr.5">locally-finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#T25" target="_self" onmouseover="rs('glib_013/T25')" onmouseout="rh()">Th25</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>

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