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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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  holds <br/><font color="Olive" title="b1">W</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#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" 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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  holds <br/><font color="Maroon" title="c2">W</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#V5" title="GLIB_001:attr.5">Path-like</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  implies <font color="Maroon" title="c2">W</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#V5" title="GLIB_001:attr.5">Path-like</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:98"/><span class="lab"><font color="Green" title="E94">A1</font></span>: 
<font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">W</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#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:98_1_1"/>
( (  not <font color="Maroon" title="c3">m</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp;  not <font color="Maroon" title="c4">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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">W</font> ) or <font color="Maroon" title="c3">m</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c4">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or  not <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or  not <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">W</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:98_1_1_1"/><span class="lab"><font color="Green" title="E95">A2</font></span>: 
(  not <font color="Maroon" title="c3">m</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp;  not <font color="Maroon" title="c4">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp; <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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">W</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">W</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#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">m9</font> = <font color="Maroon" title="c3">m</font> 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#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:98_1_1_1_1"/>
<font color="Maroon" title="c2">W</font> is  <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</a></span>;<br/><span class="kw">hence </span><a NAME="E2:98_1_1_1_1"/>
<font color="Maroon" title="c2">W</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#V4" title="GLIB_001:attr.4">Trail-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_001.html#E96"><span class="lab"><font color="Green" title="E88">Lm59</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <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#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</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">W</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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</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="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</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="Olive" title="b2">y</font> holds <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">W</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><br/><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <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#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c7">y</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">W</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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</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">y</font> implies ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">W</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><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:98_1_1_1_1"/><span class="lab"><font color="Green" title="E96">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">and </span><br/><a NAME="E4:98_1_1_1_1"/><span class="lab"><font color="Green" title="E97">A4</font></span>: 
<font color="Maroon" title="c7">y</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">W</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="E5:98_1_1_1_1"/><span class="lab"><font color="Green" title="E98">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</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">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">W</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><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">xaa1</font> = <font color="Maroon" title="c6">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a>, <a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">yaa1</font> = <font color="Maroon" title="c7">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a>, <a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a></span>;<br/><a NAME="E8:98_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <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> <font color="Maroon" title="c7">y</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:98_1_1_1_1"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="ref" href="xreal_1.html#T14" title="XREAL_1:th.14">XREAL_1:14</a></span>;<br/><a NAME="E9:98_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A6</font></span>: 
<font color="Maroon" title="c8">xaa1</font> <a href="nat_1.html#K2" title="NAT_1: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="c9">yaa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E10:98_1_1_1_1"/>
<font color="Maroon" title="c6">x</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">W</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="#E3:98_1_1_1_1"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E4:98_1_1_1_1"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:98_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</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>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E12:98_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</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> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xaa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">xaa1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_1_1_1"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="txt" href="glib_001.html#E52"><span class="lab"><font color="Green" title="E44">Lm15</font></span></a></span>;<br/><a NAME="E13:98_1_1_1_1"/><span class="lab"><font color="Green" title="E101">A8</font></span>: 
<font color="Maroon" title="c7">y</font> <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</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>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:98_1_1_1_1"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E14:98_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">yaa1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">yaa1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_1_1_1"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="txt" href="glib_001.html#E52"><span class="lab"><font color="Green" title="E44">Lm15</font></span></a></span>;<br/><a NAME="E15:98_1_1_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">yaa1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_1_1_1"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="txt" href="#E13:98_1_1_1_1"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="txt" href="glib_001.html#E52"><span class="lab"><font color="Green" title="E44">Lm15</font></span></a></span>;<br/><a NAME="E16:98_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A10</font></span>: 
<font color="Maroon" title="c5">m9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">yaa1</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">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E17:98_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A11</font></span>: 
<font color="Maroon" title="c5">m9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">yaa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E5:98_1_1_1_1"><span class="lab"><font color="Green" title="E98">A5</font></span></a>, <a class="txt" href="#E12:98_1_1_1_1"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E14:98_1_1_1_1"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="txt" href="#E9:98_1_1_1_1"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</a></span>;<br/><div><span class="lab"><font color="Green" title="E105">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:98_1_1_1_1_1"/><span class="lab"><font color="Green" title="E106">A13</font></span>: 
<font color="Maroon" title="c8">xaa1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:98_1_1_1_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_1_1_1"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E3:98_1_1_1_1_1"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">xaa1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_1_1_1_1_1"><span class="lab"><font color="Green" title="E106">A13</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E4:98_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E5:98_1_1_1_1"><span class="lab"><font color="Green" title="E98">A5</font></span></a>, <a class="txt" href="#E12:98_1_1_1_1"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E14:98_1_1_1_1"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="txt" href="#E9:98_1_1_1_1"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="txt" href="#E16:98_1_1_1_1"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</a></span>; <a class="txt" 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><a NAME="E19:98_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E5:98_1_1_1_1"><span class="lab"><font color="Green" title="E98">A5</font></span></a>, <a class="txt" href="#E12:98_1_1_1_1"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E14:98_1_1_1_1"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="txt" href="#E9:98_1_1_1_1"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="txt" href="#E16:98_1_1_1_1"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</a></span>;<br/><a NAME="E20:98_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A14</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</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>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98_1_1_1"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="txt" href="glib_001.html#E52"><span class="lab"><font color="Green" title="E44">Lm15</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E21:98_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:98_1_1_1_1"><span class="lab"><font color="Green" title="E105">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">W</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><br/><a NAME="E22:98_1_1_1_1"/>
<font color="Maroon" title="c5">m9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">xaa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E5:98_1_1_1_1"><span class="lab"><font color="Green" title="E98">A5</font></span></a>, <a class="txt" href="#E12:98_1_1_1_1"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="txt" href="#E14:98_1_1_1_1"><span class="lab"><font color="Green" title="E102">A9</font></span></a>, <a class="txt" href="#E9:98_1_1_1_1"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="txt" href="#E16:98_1_1_1_1"><span class="lab"><font color="Green" title="E103">A10</font></span></a>, <a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</a></span>;<br/><span class="kw">hence </span><a NAME="E23:98_1_1_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">W</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="#E1:98_1_1_1"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="txt" href="#E4:98_1_1_1_1"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="txt" href="#E17:98_1_1_1_1"><span class="lab"><font color="Green" title="E104">A11</font></span></a>, <a class="txt" href="#E18:98_1_1_1_1"><span class="lab"><font color="Green" title="E105">A12</font></span></a>, <a class="txt" href="#E20:98_1_1_1_1"><span class="lab"><font color="Green" title="E106">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" 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="E4:98_1_1_1"/>
<font color="Maroon" title="c2">W</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#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D28" target="_self" title="GLIB_001:def.28">Def28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:98_1_1_2"/>
( <font color="Maroon" title="c3">m</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or <font color="Maroon" title="c4">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or  not <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or  not <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">W</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">W</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#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:98_1_1_2"/>
<font color="Maroon" title="c2">W</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#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="ref" href="glib_001.html#D11" target="_self" title="GLIB_001:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:98"/>
<font color="Maroon" title="c2">W</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#V5" title="GLIB_001:attr.5">Path-like</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
