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<b>consider </b><font color="Maroon" title="c1">G</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V1" title="GLIB_003:attr.1">[Weighted]</a> <a href="glib_003.html#V2" title="GLIB_003:attr.2">[ELabeled]</a> <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a> <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> <a href="lexbfs.html#V4" title="LEXBFS:attr.4">[V2Labeled]</a> <a href="lexbfs.html#V5" title="LEXBFS:attr.5">natural-v2labeled</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/>
<b>set </b><font color="Maroon" title="c2">F</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E2:92_2_1"/><i><font color="Green" title="E42">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">F</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c1">G</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> is   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a></span><br/><a NAME="E1:92_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:92_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E2:92_2_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:92_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/><b>hence </b><a NAME="E4:92_2_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> is   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c3">F</font> as   <a href="glib_000.html#NM11" title="GLIB_000:NM.11">GraphSeq</a> <b>by </b><i><a class="ref" href="glib_000.html#D55" title="GLIB_000:def.55">GLIB_000:def 55</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_003.html#V1" title="GLIB_003:attr.1">[Weighted]</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_003.html#V2" title="GLIB_003:attr.2">[ELabeled]</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="lexbfs.html#V4" title="LEXBFS:attr.4">[V2Labeled]</a> )</span><br/><a NAME="E1:92_2_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:92_2_1_2"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E2:92_2_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:92_2_1_2"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/><b>hence </b><a NAME="E4:92_2_1_2"/>
( <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_003.html#V1" title="GLIB_003:attr.1">[Weighted]</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_003.html#V2" title="GLIB_003:attr.2">[ELabeled]</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="lexbfs.html#V4" title="LEXBFS:attr.4">[V2Labeled]</a> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">F</font> as  <a href="glib_003.html#V12" title="GLIB_003:attr.12">[Weighted]</a> <a href="glib_003.html#V13" title="GLIB_003:attr.13">[ELabeled]</a> <a href="glib_003.html#V14" title="GLIB_003:attr.14">[VLabeled]</a> <a href="lexbfs.html#V14" title="LEXBFS:attr.14">[V2Labeled]</a> <a href="glib_000.html#NM11" title="GLIB_000:NM.11">GraphSeq</a> <b>by </b><i><a class="ref" href="lexbfs.html#D20" target="_self" title="LEXBFS:def.20">Def20</a>, <a class="ref" href="glib_003.html#D24" title="GLIB_003:def.24">GLIB_003:def 24</a>, <a class="ref" href="glib_003.html#D25" title="GLIB_003:def.25">GLIB_003:def 25</a>, <a class="ref" href="glib_003.html#D26" title="GLIB_003:def.26">GLIB_003:def 26</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> is <a href="glib_000.html#V14" title="GLIB_000:attr.14">finite</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="lexbfs.html#V11" title="LEXBFS:attr.11">natural-vlabeled</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="lexbfs.html#V15" title="LEXBFS:attr.15">natural-v2labeled</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="lexbfs.html#V12" title="LEXBFS:attr.12">chordal</a> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c6">x</font> is <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> &amp; <font color="Maroon" title="c5">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c6">x</font> is <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> &amp; <font color="Maroon" title="c5">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c6">x</font> is <a href="lexbfs.html#V5" title="LEXBFS:attr.5">natural-v2labeled</a> &amp; <font color="Maroon" title="c5">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c6">x</font> is <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> )</span><br/><a NAME="E1:92_2_1_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:92_2_1_3"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E2:92_2_1"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:92_2_1_3"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/><b>hence </b><a NAME="E4:92_2_1_3"/>
( <font color="Maroon" title="c5">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c6">x</font> is <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> &amp; <font color="Maroon" title="c5">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c6">x</font> is <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> &amp; <font color="Maroon" title="c5">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c6">x</font> is <a href="lexbfs.html#V5" title="LEXBFS:attr.5">natural-v2labeled</a> &amp; <font color="Maroon" title="c5">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c6">x</font> is <a href="chord.html#V6" title="CHORD:attr.6">chordal</a> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:92_2_1"/>
( <font color="Maroon" title="c5">F</font> is <a href="glib_000.html#V14" title="GLIB_000:attr.14">finite</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="lexbfs.html#V11" title="LEXBFS:attr.11">natural-vlabeled</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="lexbfs.html#V15" title="LEXBFS:attr.15">natural-v2labeled</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="lexbfs.html#V12" title="LEXBFS:attr.12">chordal</a> )
 <b>by </b><i><a class="ref" href="lexbfs.html#D17" target="_self" title="LEXBFS:def.17">Def17</a>, <a class="ref" href="lexbfs.html#D18" target="_self" title="LEXBFS:def.18">Def18</a>, <a class="ref" href="lexbfs.html#D21" target="_self" title="LEXBFS:def.21">Def21</a>, <a class="ref" href="glib_000.html#D60" title="GLIB_000:def.60">GLIB_000:def 60</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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