<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>set </b><font color="Maroon">CSi</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">VLi</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>, <font color="Maroon">b</font> be   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>, <font color="Maroon">c</font> be   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>;<br/><b>assume </b><a NAME="E1:178_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="chord.html#R1">are_adjacent</a>  )
 ;<br/><a NAME="E2:178_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E3:178_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="glib_000.html#K40">.Lifespan()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i/>;<br/><a NAME="E4:178_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E5:178_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>reconsider </b><font color="Maroon">ag</font> = <font color="Maroon">a</font>, <font color="Maroon">bg</font> = <font color="Maroon">b</font>, <font color="Maroon">cg</font> = <font color="Maroon">c</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:178_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">d</font> being  <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">d</font> &amp; <font color="Olive">d</font>,<font color="Maroon">b</font> <a href="chord.html#R1">are_adjacent</a>  holds <br/><font color="Olive">d</font>,<font color="Maroon">a</font> <a href="chord.html#R1">are_adjacent</a> 
 ;<br/><b>reconsider </b><font color="Maroon">ai</font> = <font color="Maroon">a</font>, <font color="Maroon">bi</font> = <font color="Maroon">b</font>, <font color="Maroon">ci</font> = <font color="Maroon">c</font> as   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> ;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span>;<br/><b>set </b><font color="Maroon">CSb</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">VLb</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E3:178_1_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:178_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><b>reconsider </b><font color="Maroon">ab</font> = <font color="Maroon">a</font>, <font color="Maroon">bb</font> = <font color="Maroon">b</font> as   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>reconsider </b><font color="Maroon">cb</font> = <font color="Maroon">c</font> as   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><b>reconsider </b><font color="Maroon">sbb</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">bb</font>, <font color="Maroon">sba</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="lexbfs.html#K2" target="_self">.</a> <font color="Maroon">ab</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E8:178_1_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E9:178_1_1"/>
<font color="Maroon">sbb</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:178_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">sbb</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E11:178_1_1"/>
<font color="Maroon">sba</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E12:178_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">sba</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E13:178_1_1"/><i><font color="Green">E50</font></i>: 
( not <font color="Maroon">bb</font>,<font color="Maroon">cb</font> <a href="chord.html#R1">are_adjacent</a>  &amp; <font color="Maroon">ab</font>,<font color="Maroon">cb</font> <a href="chord.html#R1">are_adjacent</a>  )
 <b>by </b><i>, , <a class="ref" href="chord.html#T44">CHORD:44</a></i>;<br/><a NAME="E14:178_1_1"/>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">cb</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D6" target="_self">Def6</a>, <a class="ref" href="chord.html#T52">CHORD:52</a></i>;<br/><a NAME="E15:178_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">cg</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="chord.html#T51">CHORD:51</a></i>;<br/><a NAME="E16:178_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">cb</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D6" target="_self">Def6</a>, , <a class="ref" href="chord.html#T52">CHORD:52</a></i>;<br/><a NAME="E17:178_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">cg</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="chord.html#T51">CHORD:51</a></i>;<br/><a NAME="E18:178_1_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">ci</font> )
 <b>by </b><i/>;<br/><a NAME="E19:178_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">kc</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span>;<br/><b>set </b><font color="Maroon">CSc</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">VLc</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E20:178_1_1"/><i><font color="Green">E92</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  )
 <b>by </b><i>, , <a class="txt" href="lexbfs.html#E13:178_1_1"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E21:178_1_1"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E22:178_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E23:178_1_1"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E24:178_1_1"/>
<font color="Maroon">ci</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E25:178_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K10" target="_self">.PickedAt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">cb</font>
 <b>by </b><i>, </i>;<br/><a NAME="E26:178_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">cb</font> <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E27:178_1_1"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:178_1_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:178_1_1_1"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:178_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E4:178_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="lexbfs.html#E12:178_1_1"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E4:178_1_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="lexbfs.html#E35:178_1_1"><i><font color="Green">E121</font></i></a>, </i>;<br/><a NAME="E5:178_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E34:178_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="lexbfs.html#E1:178_1_1_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E6:178_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E29:178_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sba</font>
 <b>by </b><i>, , , <a class="txt" href="lexbfs.html#E32:178_1_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="lexbfs.html#E33:178_1_1"><i><font color="Green">E118</font></i></a>, </i>;<br/><a NAME="E30:178_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sba</font>
 <b>by </b><i>, , </i>;<br/><div><i><font color="Green">E102</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:178_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sbb</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:178_1_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:178_1_1_2"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font>
 <b>and </b><br/><a NAME="E5:178_1_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">bb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E23:178_1_1"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E6:178_1_1_2"/>
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:178_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><a NAME="E32:178_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> 
 <b>by </b><i><a class="txt" href="lexbfs.html#E34:178_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="lexbfs.html#E35:178_1_1"><i><font color="Green">E121</font></i></a>, </i>;<br/><a NAME="E33:178_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a>  <a href="termord.html#R1">&lt;=</a> <font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> , <a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="txt" href="lexbfs.html#E22:178_1_1"><i><font color="Green">E94</font></i></a></i>;<br/><b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font>;<br/><a NAME="E34:178_1_1"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="lexbfs.html#E35:178_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="uproots.html#T8">UPROOTS:8</a></i>;<br/><a NAME="E35:178_1_1"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="lexbfs.html#E34:178_1_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="uproots.html#T9">UPROOTS:9</a></i>;<br/><a NAME="E36:178_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:178_1_1_3_1"/>
( for <font color="Olive">k</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> or  ex <font color="Olive">k</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; not <span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:178_1_1_3_1_1"/>
for <font color="Olive">k</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:178_1_1_3_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="bagorder.html#D8">BAGORDER:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:178_1_1_3_1_2"/>
 ex <font color="Olive">k</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; not <span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">k</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">qq</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:178_1_1_3_1_2"/><i><font color="Green">E122</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">qq</font> &amp; <font color="Maroon">qq</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>and </b><br/><a NAME="E4:178_1_1_3_1_2"/><i><font color="Green">E327</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">qq</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">qq</font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">qq</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="lexbfs.html#E18"><i><font color="Green">Lemma126</font></i></a></i>;<br/><a NAME="E6:178_1_1_3_1_2"/><i><font color="Green">E330</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i><a class="txt" href="lexbfs.html#E18"><i><font color="Green">Lemma126</font></i></a>, <a class="txt" href="lexbfs.html#E19"><i><font color="Green">Lemma127</font></i></a></i>;<br/><a NAME="E7:178_1_1_3_1_2"/><i><font color="Green">E331</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1_1_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:178_1_1_3_1_2_1"/><i><font color="Green">E332</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>consider </b><font color="Maroon">ok</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:178_1_1_3_1_2_1"/><i><font color="Green">E334</font></i>: 
<font color="Maroon">ok</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E4:178_1_1_3_1_2_1"/>
( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ok</font> &amp; <font color="Maroon">ok</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>and </b><br/><a NAME="E5:178_1_1_3_1_2_1"/><i><font color="Green">E335</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">ok</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">ok</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:178_1_1_3_1_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sba</font> or  not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sba</font> )
 
;<br/>
<a NAME="E7:178_1_1_3_1_2_1"/><b>then </b><i><font color="Green">E336</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="uproots.html#T8">UPROOTS:8</a>, <a class="ref" href="uproots.html#T9">UPROOTS:9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_1_1_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:178_1_1_3_1_2_1_1"/>
(  not <font color="Maroon">ok</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sbb</font> or <font color="Maroon">ok</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sbb</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_1_1_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:178_1_1_3_1_2_1_1_1"/>
not <font color="Maroon">ok</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sbb</font>
 ;<br/><div class="add"><a NAME="E2:178_1_1_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">ok</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sba</font>
 <b>by </b><i>, , , <a class="ref" href="uproots.html#T8">UPROOTS:8</a></i>;<br/><b>hence </b><a NAME="E3:178_1_1_3_1_2_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_1_1_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:178_1_1_3_1_2_1_1_2"/>
<font color="Maroon">ok</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sbb</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:178_1_1_3_1_2_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">mm</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:178_1_1_3_1_2"/><i><font color="Green">E338</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">mm</font>]
 <b>and </b><br/><a NAME="E10:178_1_1_3_1_2"/><i><font color="Green">E339</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mm</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, <a class="txt" href="lexbfs.html#E20"><i><font color="Green">Lemma128</font></i></a>);<br/></i><a NAME="E11:178_1_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">mm</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/><b>then consider </b><font color="Maroon">jy</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:178_1_1_3_1_2"/><i><font color="Green">E341</font></i>: 
( <font color="Maroon">jy</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> &amp; <font color="Maroon">jy</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">mm</font> )
 ;<br/><a NAME="E14:178_1_1_3_1_2"/>
( <font color="Maroon">mm</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sbb</font> or  not <font color="Maroon">mm</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sbb</font> )
 ;<br/><a NAME="E15:178_1_1_3_1_2"/><b>then </b><i><font color="Green">E342</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">mm</font> <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">mm</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="uproots.html#T8">UPROOTS:8</a>, <a class="ref" href="uproots.html#T9">UPROOTS:9</a></i>;<br/><div><i><font color="Green">E343</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1_1_3_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:178_1_1_3_1_2_2"/><i><font color="Green">E344</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">mm</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:178_1_1_3_1_2_2"/><b>then </b>
<font color="Maroon">mm</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sbb</font>
 <b>by </b><i><a class="ref" href="uproots.html#T8">UPROOTS:8</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:178_1_1_3_1_2_2"/><i><font color="Green">E345</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:178_1_1_3_1_2_2"/><i><font color="Green">E346</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</font>
 <b>and </b><br/><a NAME="E6:178_1_1_3_1_2_2"/><i><font color="Green">E347</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E23:178_1_1"><i><font color="Green">E95</font></i></a></i>;<br/><a NAME="E7:178_1_1_3_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E8:178_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E348</font></i>: 
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E2"><i><font color="Green">Lemma52</font></i></a></i>;<br/><a NAME="E9:178_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E349</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/><a NAME="E10:178_1_1_3_1_2_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a></i>;<br/><b>then reconsider </b><font color="Maroon">zi</font> = <font color="Maroon">z</font> as   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <b>by </b><i><a class="txt" href="lexbfs.html#E30:178_1_1"><i><font color="Green">E101</font></i></a></i>;<br/><a NAME="E12:178_1_1_3_1_2_2"/><i><font color="Green">E351</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">mm</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E13:178_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E352</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</font>
 <b>by </b><i>, , <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E14:178_1_1_3_1_2_2"/><i><font color="Green">E353</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">cb</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E15:178_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E354</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E16:178_1_1_3_1_2_2"/><i><font color="Green">E355</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">zi</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T11" target="_self">Th11</a>, , , <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a>, , </i>;<br/><b>reconsider </b><font color="Maroon">zg</font> = <font color="Maroon">z</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E18:178_1_1_3_1_2_2"/>
<font color="Maroon">bb</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , <a class="ref" href="chord.html#T51">CHORD:51</a></i>;<br/><a NAME="E19:178_1_1_3_1_2_2"/><b>then </b>
<font color="Maroon">bb</font>,<font color="Maroon">z</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i><a class="ref" href="chord.html#T52">CHORD:52</a></i>;<br/><a NAME="E20:178_1_1_3_1_2_2"/><b>then </b>
<font color="Maroon">zi</font>,<font color="Maroon">bi</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T44">CHORD:44</a></i>;<br/><a NAME="E21:178_1_1_3_1_2_2"/><b>then </b>
<font color="Maroon">zi</font>,<font color="Maroon">ai</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, , , , <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E22:178_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E357</font></i>: 
<font color="Maroon">z</font>,<font color="Maroon">ab</font> <a href="chord.html#R1">are_adjacent</a> 
 <b>by </b><i>, <a class="ref" href="chord.html#T44">CHORD:44</a></i>;<br/><a NAME="E23:178_1_1_3_1_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">zg</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E31:178_1_1"><i><font color="Green">E102</font></i></a>, , , <a class="ref" href="chord.html#T52">CHORD:52</a></i>;<br/><a NAME="E24:178_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E358</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">zg</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="chord.html#T51">CHORD:51</a></i>;<br/><b>set </b><font color="Maroon">kc</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span>;<br/><b>set </b><font color="Maroon">CSc</font> = <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">VLc</font> =  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E25:178_1_1_3_1_2_2"/><i><font color="Green">E359</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a>, <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, </i>;<br/><a NAME="E26:178_1_1_3_1_2_2"/><i><font color="Green">E360</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, , <a class="ref" href="lexbfs.html#T12" target="_self">Th12</a>, </i>;<br/><a NAME="E27:178_1_1_3_1_2_2"/><i><font color="Green">E361</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E28:178_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E362</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a>  )
 <b>by </b><i>, <a class="txt" href="lexbfs.html#E13:178_1_1"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E29:178_1_1_3_1_2_2"/><i><font color="Green">E363</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="lexbfs.html#E13:178_1_1"><i><font color="Green">E50</font></i></a></i>;<br/><a NAME="E30:178_1_1_3_1_2_2"/><i><font color="Green">E364</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#T13" target="_self">Th13</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E31:178_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E365</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><a NAME="E32:178_1_1_3_1_2_2"/><i><font color="Green">E366</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#T13" target="_self">Th13</a>, <a class="ref" href="lexbfs.html#D13" target="_self">Def13</a>, </i>;<br/><a NAME="E33:178_1_1_3_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K10" target="_self">.PickedAt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E34:178_1_1_3_1_2_2"/><b>then </b><i><font color="Green">E367</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="lexbfs.html#K13" target="_self">LexBFS:PickUnnumbered</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E35:178_1_1_3_1_2_2"/><i><font color="Green">E368</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1_1_3_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:178_1_1_3_1_2_2_1"/><i><font color="Green">E369</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:178_1_1_3_1_2_2_1"/><b>then </b><i><font color="Green">E370</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:178_1_1_3_1_2_2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="lexbfs.html#E12:178_1_1"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E4:178_1_1_3_1_2_2_1"/><b>then </b><i><font color="Green">E371</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D14" target="_self">Def14</a>, , </i>;<br/><a NAME="E5:178_1_1_3_1_2_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:178_1_1_3_1_2_2_1"/><b>then </b><i><font color="Green">E372</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D12" target="_self">Def12</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E7:178_1_1_3_1_2_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:178_1_1_3_1_2_2_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><a NAME="E37:178_1_1_3_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">sba</font>
 <b>by </b><i>, , , <a class="ref" href="lexbfs.html#D11" target="_self">Def11</a>, </i>;<br/><b>hence </b><a NAME="E38:178_1_1_3_1_2_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="lexbfs.html#D14" target="_self">Def14</a>, , <a class="ref" href="uproots.html#T9">UPROOTS:9</a></i>;<br/></div><b>end;</b></div><a NAME="E17:178_1_1_3_1_2"/><i><font color="Green">E373</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">mm</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">mm</font>
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1_1_3_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:178_1_1_3_1_2_3"/><i><font color="Green">E374</font></i>: 
( <font color="Maroon">mm</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 ;<br/><b>reconsider </b><font color="Maroon">kk</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a></i>;<br/><b>assume </b><a NAME="E3:178_1_1_3_1_2_3"/><i><font color="Green">E375</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E4:178_1_1_3_1_2_3"/><i><font color="Green">E376</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">bi</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">cb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, , <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><a NAME="E5:178_1_1_3_1_2_3"/>
<font color="Maroon">mm</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">kk</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="lexbfs.html#D7" target="_self">Def7</a>, <a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/><b>then consider </b><font color="Maroon">mmy</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:178_1_1_3_1_2_3"/><i><font color="Green">E378</font></i>: 
( <font color="Maroon">mmy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</font> &amp; <font color="Maroon">mmy</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">kk</font> )
 ;<br/><b>thus </b><a NAME="E8:178_1_1_3_1_2_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="lexbfs.html#D9" target="_self">Def9</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:178_1_1_3_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="bagorder.html#D8">BAGORDER:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E37:178_1_1"/><b>then </b>
<font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a>  <a href="termord.html#R1">&lt;=</a> <font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> , <a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="termord.html#D2">TERMORD:def 2</a></i>;<br/><a NAME="E38:178_1_1"/><b>then </b><i><font color="Green">E379</font></i>: 
<font color="Maroon">sbb</font>,1 <a href="uproots.html#K2">-bag</a>  <a href="termord.html#R2">&lt;</a> <font color="Maroon">sba</font>,1 <a href="uproots.html#K2">-bag</a> , <a href="bagorder.html#K4">InvLexOrder</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="termord.html#D3">TERMORD:def 3</a></i>;<br/><b>thus </b><a NAME="E39:178_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lexbfs.html#E18"><i><font color="Green">Lemma126</font></i></a>, , <a class="ref" href="termord.html#T5">TERMORD:5</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">x</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E9:178_1"/><i><font color="Green">E380</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">b</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">x</font>,<font color="Maroon">a</font> <a href="chord.html#R1">are_adjacent</a>  )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">v</font> being  <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> st <br/>( <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font>,<font color="Olive">v</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">a</font>,<font color="Olive">v</font> <a href="chord.html#R1">are_adjacent</a>  &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">v</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E10:178_1"/><i><font color="Green">E381</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i/>;<br/><a NAME="E11:178_1"/><i><font color="Green">E382</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:178_1_2"/><i><font color="Green">E383</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>consider </b><font color="Maroon">v</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E3:178_1_2"/><i><font color="Green">E384</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <font color="Maroon">v</font>,<font color="Maroon">b</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">v</font>,<font color="Maroon">a</font> <a href="chord.html#R1">are_adjacent</a>  )
 <b>and </b><br/><a NAME="E4:178_1_2"/><i><font color="Green">E385</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:178_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:178_1_2"/><b>then </b><i><font color="Green">E386</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:178_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>hence </b><a NAME="E8:178_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K16">.order()</a> 
 <b>by </b><i><a class="ref" href="lexbfs.html#D6" target="_self">Def6</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:178_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, );<br/></i><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:178_1"/><i><font color="Green">E387</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E15:178_1"/><i><font color="Green">E388</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><b>consider </b><font color="Maroon">v</font> being   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E17:178_1"/><i><font color="Green">E389</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font>,<font color="Maroon">v</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">a</font>,<font color="Maroon">v</font> <a href="chord.html#R1">are_adjacent</a>  )
 <b>and </b><br/><a NAME="E18:178_1"/><i><font color="Green">E390</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E19:178_1"/><i><font color="Green">E391</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E20:178_1"/>
for <font color="Olive">d</font> being  <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>  st <font color="Olive">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> &amp; <font color="Olive">d</font>,<font color="Maroon">b</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Olive">d</font>,<font color="Maroon">a</font> <a href="chord.html#R1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">d</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:178_1_3"/><i><font color="Green">E392</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">v</font> &amp; <font color="Maroon">d</font>,<font color="Maroon">b</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">d</font>,<font color="Maroon">a</font> <a href="chord.html#R1">are_adjacent</a>  )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:178_1_3_1"/>
( <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> or <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_1_3_1_1"><b>suppose </b></a><a NAME="E1:178_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:178_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#D6" target="_self">Def6</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="178_1_3_1_2"><b>suppose </b></a><a NAME="E1:178_1_3_1_2"/><i><font color="Green">E393</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font>
 ;<br/><div class="add"><div><i><font color="Green">E394</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="178_1_3_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:178_1_3_1_2_1"/>
not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:178_1_3_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:178_1_3_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:178_1_3_1_2"/><i><font color="Green">E395</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E4:178_1_3_1_2"/>
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i/>;<br/><a NAME="E5:178_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E6:178_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">d</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E21:178_1"/>
 ex <font color="Olive">d</font> being  <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span> st <br/>( <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">d</font> &amp; <font color="Maroon">b</font>,<font color="Olive">d</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Maroon">a</font>,<font color="Olive">d</font> <a href="chord.html#R1">are_adjacent</a>  &amp; ( for <font color="Olive">e</font> being  <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span>  st <font color="Olive">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> &amp; <font color="Olive">e</font>,<font color="Maroon">b</font> <a href="chord.html#R1">are_adjacent</a>  &amp; not <font color="Olive">e</font>,<font color="Maroon">a</font> <a href="chord.html#R1">are_adjacent</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">e</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">d</font> ) )
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:178"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K18" target="_self">LexBFS:CSeq</a> <font color="Maroon">G</font></span>)</span> <a href="lexbfs.html#K9" target="_self">.</a> <font color="Maroon">i</font> is <a href="lexbfs.html#V21" target="_self">with_property_L3</a>
 <b>by </b><i/>;<br/>


</div>
