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<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="E1:127_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="127_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_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="E1:127_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="127_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:127_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>then 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>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">a</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:127_1_1_1_1_1_1"/><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</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">a</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">b</font> )
 ;<br/><a NAME="E2:127_1_1_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:127_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</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">b</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_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="E1:127_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">a</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">a</font> &amp; <font color="Maroon">a</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>, <a class="ref" href="glib_001.html#D11" target="_self">Def11</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">a</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="E2:127_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E44</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">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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="127_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="127_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="E1:127_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">b</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">b</font> &amp; <font color="Maroon">b</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>, <a class="ref" href="glib_001.html#D11" target="_self">Def11</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">b</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="E2:127_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E45</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">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">b</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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:127_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</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"><b>then reconsider </b><font color="Maroon">b2</font> = <span class="p1">(<span class="default"><font color="Maroon">b</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">even</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="E3:127_1_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E47</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">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">b2</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_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="127_1_1_1_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="E1:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</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">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E4:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E5:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E6:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/>
contradiction
 <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="E5:127_1_1_1_1_1_1_1_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:127_1_1_1_1_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:127_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">a</font> &amp; <font color="Maroon">a</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"><b>reconsider </b><font color="Maroon">a2</font> = <span class="p1">(<span class="default"><font color="Maroon">a</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">even</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/><b>reconsider </b><font color="Maroon">nm4</font> = <font color="Maroon">n'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m'</font> as  <a href="abian.html#V1">even</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="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E4:127_1_1_1_1_1_1_1_1_2"/><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">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_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="127_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="E1:127_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">b</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">b</font> &amp; <font color="Maroon">b</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>, <a class="ref" href="glib_001.html#D11" target="_self">Def11</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">b</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="E2:127_1_1_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">b</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="E3:127_1_1_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 <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="127_1_1_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:127_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">b</font> &amp; <font color="Maroon">b</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"><b>reconsider </b><font color="Maroon">b2</font> = <span class="p1">(<span class="default"><font color="Maroon">b</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">even</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="E3:127_1_1_1_1_1_1_1_1_2_1_1_2"/><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">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">b2</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_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="E1:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font> or <font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
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/><a NAME="E3:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">nm4</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E5:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">b2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font>
 ;<br/><div class="add"><a NAME="E2:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">nm4</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">nm4</font>
 ;<br/><b>hence </b><a NAME="E3:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:127_1_1_1_1_1_1_1_1_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:127_1_1_1_1_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:127_1_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:127_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#V4" target="_self">Trail-like</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:127_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="E2:127_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#V4" target="_self">Trail-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="E3:127_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#V4" target="_self">Trail-like</a>
 ;<br/>


</div>
