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<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E1:127_1_1"/><i><font color="Green">E104</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="glib_001.html#E67"><i><font color="Green">Lemma59</font></i></a></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">c<sub>3</sub></font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">c<sub>4</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> ) or <font color="Maroon">c<sub>3</sub></font> is <a href="abian.html#V1">even</a> or <font color="Maroon">c<sub>4</sub></font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> or  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></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">E105</font></i>: 
( not <font color="Maroon">c<sub>3</sub></font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">c<sub>4</sub></font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">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">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:127_1_1_1_1_1_1"/><i><font color="Green">E106</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/><a NAME="E2:127_1_1_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <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">E107</font></i>: 
( <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <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">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E3:127_1_1_1_1_1_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="glib_001.html#E75"><i><font color="Green">Lemma67</font></i></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">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></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">E108</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E70"><i><font color="Green">Lemma62</font></i></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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E3:127_1_1_1_1_1_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="glib_001.html#E75"><i><font color="Green">Lemma67</font></i></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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></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">E109</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E70"><i><font color="Green">Lemma62</font></i></a></i>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <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><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="glib_001.html#E99"><i><font color="Green">Lemma91</font></i></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_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">E109</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green">Lemma63</font></i></a></i>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E110</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green">Lemma63</font></i></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></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><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="glib_001.html#E3:127_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="glib_001.html#E99"><i><font color="Green">Lemma91</font></i></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_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">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></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">E111</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>9</sub></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">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <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">E112</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green">E111</font></i></a></i>;<br/><a NAME="E5:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <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><a class="txt" href="glib_001.html#E4:127_1_1_1_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green">E112</font></i></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_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">E108</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green">Lemma63</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <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">E109</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green">Lemma63</font></i></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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E3:127_1_1_1_1_1_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="glib_001.html#E75"><i><font color="Green">Lemma67</font></i></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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></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">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></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="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <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">E110</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green">Lemma63</font></i></a></i>;<br/><a NAME="E3:127_1_1_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E111</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="glib_001.html#E71"><i><font color="Green">Lemma63</font></i></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">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></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">E112</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></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">c<sub>6</sub></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">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2"><i><font color="Green">E108</font></i></a>, <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">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></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><a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_001.html#E4:127_1_1_1_1_1_1_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="glib_001.html#E3:127_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="glib_001.html#E1:127_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="glib_001.html#E99"><i><font color="Green">Lemma91</font></i></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_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">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></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">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></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="txt" href="glib_001.html#E1:127_1_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="glib_001.html#V4" target="_self">Trail-like</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E99"><i><font color="Green">Lemma91</font></i></a></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">c<sub>3</sub></font> is <a href="abian.html#V1">even</a> or <font color="Maroon">c<sub>4</sub></font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> or  not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:127_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="glib_001.html#V4" target="_self">Trail-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self">Def12</a></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">c<sub>2</sub></font> <a href="glib_001.html#K9" target="_self">.remove</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="glib_001.html#V4" target="_self">Trail-like</a>
 ;<br/>


</div>
