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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">P</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  holds <br/> for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">P</font> holds <br/>( <font color="Olive" title="b1">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">n</font>) is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">n</font>) is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  &amp; <font color="Maroon" title="c2">P</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  implies  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> holds <br/>( <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>) is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>) is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:104"/><span class="lab"><font color="Green" title="E84">A1</font></span>: 
<font color="Maroon" title="c2">P</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> 
 <span class="kw">and </span><br/><a NAME="E2:104"/><span class="lab"><font color="Green" title="E85">A2</font></span>: 
<font color="Maroon" title="c2">P</font> is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> holds <br/>( <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>) is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font>) is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font> implies ( <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>) is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>) is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:104"/><span class="lab"><font color="Green" title="E86">A3</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">and </span><br/><a NAME="E4:104"/><span class="lab"><font color="Green" title="E87">A4</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>) is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>) is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">Q</font> = <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>);<br/>
<a NAME="E5:104"/><span class="lab"><font color="Green" title="E88">A5</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E6:104"/><span class="lab"><font color="Green" title="E89">A6</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>) is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:104_1"/>
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>) is  <a href="chord.html#V5" title="CHORD:attr.5">chordal</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:104_1"/><span class="lab"><font color="Green" title="E90">A7</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">and </span><br/><a NAME="E4:104_1"/><span class="lab"><font color="Green" title="E91">A8</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:104_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">and </span><br/><a NAME="E6:104_1"/><span class="lab"><font color="Green" title="E92">A9</font></span>: 
 ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E7:104_1"/>
 for <font color="Olive" title="b1">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  holds <br/> not <font color="Olive" title="b1">f</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c8">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:104_1"/><span class="lab"><font color="Green" title="E93">A10</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:104_1"><span class="lab"><font color="Green" title="E92">A9</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">mi</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/><span class="kw">set </span><font color="Maroon" title="c10">mj</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/><a NAME="E10:104_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T2" target="_self" onmouseover="rs('chord/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E11:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A11</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:104_1"><span class="lab"><font color="Green" title="E91">A8</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E12:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E6:104"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E5:104"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="ref" href="glib_001.html#T47" onmouseover="rs('glib_001/T47')" onmouseout="rh()">GLIB_001:47</a></span>;<br/><a NAME="E13:104_1"/><span class="lab"><font color="Green" title="E96">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E6:104"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E5:104"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="txt" href="#E11:104_1"><span class="lab"><font color="Green" title="E94">A11</font></span></a>, <a class="ref" href="glib_001.html#T47" onmouseover="rs('glib_001/T47')" onmouseout="rh()">GLIB_001:47</a></span>;<br/><a NAME="E14:104_1"/>
<font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E15:104_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104_1"><span class="lab"><font color="Green" title="E90">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E16:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A14</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:104_1"><span class="lab"><font color="Green" title="E91">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E17:104_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T2" target="_self" onmouseover="rs('chord/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E18:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A15</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:104_1"><span class="lab"><font color="Green" title="E97">A14</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E19:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E6:104"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E5:104"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="ref" href="glib_001.html#T47" onmouseover="rs('glib_001/T47')" onmouseout="rh()">GLIB_001:47</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">mj</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:104_1"><span class="lab"><font color="Green" title="E96">A13</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">mj</font> = <font color="Maroon" title="c11">mj</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E22:104_1"/><span class="lab"><font color="Green" title="E100">A17</font></span>: 
<font color="Maroon" title="c12">mj</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:104_1"><span class="lab"><font color="Green" title="E96">A13</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">mi</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:104_1"><span class="lab"><font color="Green" title="E99">A16</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">mi</font> = <font color="Maroon" title="c13">mi</font> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E25:104_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104_1"><span class="lab"><font color="Green" title="E90">A7</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E26:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A18</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E27:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A19</font></span>: 
<font color="Maroon" title="c14">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">mj</font>
 ;<br/><a NAME="E28:104_1"/>
<font color="Maroon" title="c14">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">mi</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E29:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A20</font></span>: 
<font color="Maroon" title="c14">mi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">mj</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:104_1"><span class="lab"><font color="Green" title="E101">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E30:104_1"/>
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">mi</font>,<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">mj</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E6:104"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E5:104"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="txt" href="#E9:104_1"><span class="lab"><font color="Green" title="E93">A10</font></span></a>, <a class="txt" href="#E18:104_1"><span class="lab"><font color="Green" title="E98">A15</font></span></a>, <a class="txt" href="#E12:104_1"><span class="lab"><font color="Green" title="E95">A12</font></span></a>, <a class="ref" href="glib_001.html#T47" onmouseover="rs('glib_001/T47')" onmouseout="rh()">GLIB_001:47</a></span>;<br/><span class="kw">hence </span><a NAME="E31:104_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E2:104"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E27:104_1"><span class="lab"><font color="Green" title="E102">A19</font></span></a>, <a class="txt" href="#E29:104_1"><span class="lab"><font color="Green" title="E103">A20</font></span></a>, <a class="txt" href="#E22:104_1"><span class="lab"><font color="Green" title="E100">A17</font></span></a>, <a class="ref" href="chord.html#T92" target="_self" onmouseover="rs('chord/T92')" onmouseout="rh()">Th91</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:104"/>
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>) is  <a href="chord.html#NV6" title="CHORD:NV.6">chordless</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>) is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>) is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:104_2"/>
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>) is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:104_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A21</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ;<br/><a NAME="E3:104_2"/><span class="lab"><font color="Green" title="E91">A22</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E6:104"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E5:104"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="ref" href="glib_001.html#T37" onmouseover="rs('glib_001/T37')" onmouseout="rh()">GLIB_001:37</a></span>;<br/><a NAME="E4:104_2"/>
<font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E6:104"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E5:104"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="ref" href="glib_001.html#T37" onmouseover="rs('glib_001/T37')" onmouseout="rh()">GLIB_001:37</a></span>;<br/><span class="kw">hence </span><a NAME="E5:104_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="txt" href="#E4:104"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="txt" href="#E6:104"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E5:104"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="txt" href="#E2:104_2"><span class="lab"><font color="Green" title="E90">A21</font></span></a>, <a class="txt" href="#E3:104_2"><span class="lab"><font color="Green" title="E91">A22</font></span></a>, <a class="ref" href="glib_001.html#T147" onmouseover="rs('glib_001/T147')" onmouseout="rh()">GLIB_001:147</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:104"/>
<font color="Maroon" title="c2">P</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> (<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>) is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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