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

<b>set </b><font color="Maroon">W2</font> = <font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>;<br/>
<a NAME="E1:128_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:128_1_1_1_1"/>
( ( not <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> ) or <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> or <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> or  not <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:128_1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
( not <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m</font>, <font color="Maroon">n'</font> = <font color="Maroon">n</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">y</font> be  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:128_1_1_1_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 ;<br/><a NAME="E2:128_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:128_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:128_1_1_1_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D11" target="_self">Def11</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:128_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> ) )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:128_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:128_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T5" target="_self">Th5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:128_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> ) )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:128_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:128_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E3:128_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E4:128_1_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:128_1_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:128_1_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T6" target="_self">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon">y2</font> = <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n'</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, , <a class="ref" href="glib_001.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:128_1_1_1_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E5:128_1_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T6" target="_self">Th6</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:128_1_1_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i>, , , </i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="glib_001.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>

;<br/><b>hence </b><a NAME="E8:128_1_1_1_1_1_1_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:128_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:128_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:128_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T6" target="_self">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon">x2</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n'</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, , <a class="ref" href="glib_001.html#T6" target="_self">Th6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1_1_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1_1_1_1_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="E1:128_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> ) )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:128_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:128_1_1_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E3:128_1_1_1_1_1_1_1_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:128_1_1_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:128_1_1_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T6" target="_self">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon">y2</font> = <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n'</font> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, , <a class="ref" href="glib_001.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:128_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E5:128_1_1_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y2</font>
 ;<br/><a NAME="E6:128_1_1_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 <b>by </b><i>, , , </i>;<br/><b>then </b><i><font color="Green">E90</font></i>: <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="glib_001.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>

;<br/><b>reconsider </b><font color="Maroon">xm4</font> = <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</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="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E9:128_1_1_1_1_1_1_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n'</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E10:128_1_1_1_1_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n'</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E11:128_1_1_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">xm4</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E12:128_1_1_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E13:128_1_1_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, , , , , , <a class="ref" href="glib_001.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E14:128_1_1_1_1_1_1_1_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m'</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><b>hence </b><a NAME="E15:128_1_1_1_1_1_1_1_1_2_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self">Def12</a>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:128_1_1_1_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:128_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:128_1_1_1_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="128_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:128_1_1_1_1_2"/>
( <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> or <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> or  not <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:128_1_1_1_1_2"/>
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:128_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> is <a href="glib_001.html#V5" target="_self">Path-like</a>
 ;<br/>


</div>
