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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#K8" target="_self">.cut</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_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="E1:123_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> ) 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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:123_1_1_1_1_1"/><i><font color="Green">E104</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> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</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="123_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:123_1_1_1_1_1_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>7</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#K8" target="_self">.cut</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>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#K8" target="_self">.cut</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#K8" target="_self">.cut</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>7</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#K8" target="_self">.cut</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> )
 ;<br/><a NAME="E2:123_1_1_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> 1, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">even</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E4:123_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:123_1_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="E5:123_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:123_1_1_1_1_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="glib_001.html#E56"><i><font color="Green">Lemma48</font></i></a></i>;<br/><a NAME="E6:123_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <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> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <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="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:123_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="glib_001.html#E5:123_1_1_1_1_1_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="glib_001.html#E1:123_1_1_1_1_1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="glib_001.html#D29" target="_self">Def29</a></i>;<br/><a NAME="E8:123_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>hence </b><a NAME="E9:123_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:123_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="glib_001.html#V6" target="_self">vertex-distinct</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self">Def29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:123_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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:123_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="glib_001.html#V6" target="_self">vertex-distinct</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D11" target="_self">Def11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:123_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="glib_001.html#K8" target="_self">.cut</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="glib_001.html#V6" target="_self">vertex-distinct</a>
 ;<br/>


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