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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green">E19</font></i>: 
( not <font color="Maroon">c<sub>3</sub></font> is <a href="graph_4.html#V1" target="_self">Simple</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31"/><i><font color="Green">E20</font></i>: 
( 1 <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#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</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>4</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>5</sub></font> &amp; ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="graph_4.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E4:31"/><i><font color="Green">E21</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E7:31"/><i><font color="Green">E23</font></i>: 
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:31"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<i><font color="Green">E25</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="graph_4.html#E7:31"><i><font color="Green">E23</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>

;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1"><b>suppose </b></a><a NAME="E1:31_2_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><a NAME="E2:31_2_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:31_2_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:31_2_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:31_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:31_2_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:31_2_1"/><i><font color="Green">E29</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:31_2_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:31_2_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:31_2_1"/><i><font color="Green">E32</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = 1,<font color="Maroon">c<sub>6</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="graph_4.html#T13" target="_self">Th13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="graph_4.html#T13" target="_self">Th13</a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = 1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = 1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E15:31_2_1"/><i><font color="Green">E33</font></i>: 
( 1 <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> )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:31_2_1"/><i><font color="Green">E34</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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="graph_4.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self">Def5</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:31_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="txt" href="graph_4.html#E9:31"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E18:31_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E19:31_2_1"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E17:31_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E20:31_2_1"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E18:31_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E21:31_2_1"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E22:31_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E24:31_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E25:31_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><b>then </b><i><font color="Green">E40</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1

;<br/><a NAME="E27:31_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E28:31_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E30:31_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><a NAME="E31:31_2_1"/><i><font color="Green">E43</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_1"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E32:31_2_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E45</font></i>: <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="graph_2.html#T7">GRAPH_2:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>,1 <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E32:31_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="graph_2.html#T8">GRAPH_2:8</a></i>
;<br/><a NAME="E34:31_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E24:31_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E35:31_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="ref" href="graph_2.html#T8">GRAPH_2:8</a></i>;<br/>1,1 <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="graph_4.html#E32:31_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="graph_2.html#T6">GRAPH_2:6</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="graph_4.html#E31:31_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="graph_2.html#T6">GRAPH_2:6</a></i>
;<br/><a NAME="E37:31_2_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E33:31_2_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="graph_4.html#E35:31_2_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E39:31_2_1"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E40:31_2_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span>  holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E41:31_2_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_1_4"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_4"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) )
 ;<br/>
<a NAME="E4:31_2_1_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_1_4"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E43:31_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_1_5"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_5"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_1_5"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_5"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_5"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:31_2_1_5"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E7:31_2_1_5"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_5"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_2_1_5"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_5"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:31_2_1_5"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E40:31_2_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_5"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_5"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_2_1_5"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_5"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_5"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_5"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>19</sub></font>
;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E46:31_2_1"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 <b>and </b><br/><a NAME="E47:31_2_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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>}</span>  holds <br/><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E48:31_2_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_1_6"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_6"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</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/>
<a NAME="E4:31_2_1_6"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E5:31_2_1_6"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</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="graph_4.html#E15:31_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_6"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:31_2_1_6"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_6"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E50:31_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_1_7"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_7"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_1_7"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_7"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_7"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:31_2_1_7"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</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="graph_4.html#E46:31_2_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E7:31_2_1_7"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E8:31_2_1_7"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</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="graph_4.html#E15:31_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_7"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:31_2_1_7"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_7"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:31_2_1_7"/>
<font color="Maroon">c<sub>25</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>26</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_4.html#E47:31_2_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_7"><i><font color="Green">E56</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_7"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:31_2_1_7"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_7"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_7"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_1_7"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/><a NAME="E52:31_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 1 <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>5</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <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> )
 <b>by </b><i><a class="txt" href="graph_4.html#E15:31_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>8</sub></font> as  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E17:31_2_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="graph_4.html#E18:31_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="graph_4.html#T15" target="_self">Th15</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>24</sub></font>
;<br/><b>take </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>13</sub></font>;<br/><a NAME="E54:31_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E37:31_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="graph_2.html#D2">GRAPH_2:def 2</a></i>;<br/><a NAME="E55:31_2_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>;<br/><a NAME="E56:31_2_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E57:31_2_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E58:31_2_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="graph_4.html#E5:31"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E55:31_2_1"><i><font color="Green">E56</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E60:31_2_1"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E58:31_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/><b>thus </b><a NAME="E61:31_2_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E17:31_2_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="graph_4.html#E18:31_2_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_4.html#E52:31_2_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="graph_4.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E62:31_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="graph_4.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E63:31_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E60:31_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E64:31_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E65:31_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E17:31_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E66:31_2_1"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/><b>hence </b><a NAME="E67:31_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E15:31_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>;<br/><a NAME="E68:31_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>;<br/><a NAME="E69:31_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E70:31_2_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E71:31_2_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E72:31_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E68:31_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="graph_2.html#T16">GRAPH_2:16</a></i>;<br/><b>set </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_9_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_9_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) )
 ;<br/><a NAME="E4:31_2_1_9_1"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  or <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_9_1"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_9_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_9"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_9_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_9_2"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  or <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_9"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_9_2_1"><b>suppose </b></a><a NAME="E1:31_2_1_9_2_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_9_2_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:31_2_1_9_2_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_9_2_1"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_9_2_2"><b>suppose </b></a><a NAME="E1:31_2_1_9_2_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_9_2_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:31_2_1_9_2_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_9_2_2"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E74:31_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E75:31_2_1"/><i><font color="Green">E62</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_10"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_2_1_10_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_10_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E4:31_2_1_10_1"/>
<font color="Maroon">c<sub>29</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_10_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_10_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_10_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:31_2_1_10"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:31_2_1_10"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<a NAME="E5:31_2_1_10"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_10"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:31_2_1_10"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_10"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="graph_4.html#E75:31_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>30</sub></font>, <font color="Maroon">c<sub>31</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:31_2_1_11"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>32</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_11"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><b>assume </b><a NAME="E4:31_2_1_11"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:31_2_1_11"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E7:31_2_1_11"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E9:31"><i><font color="Green">E25</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:31_2_1_11"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_11"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E9:31_2_1_11"/>
<font color="Maroon">c<sub>30</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_11"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_11"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E79:31_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E75:31_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/><a NAME="E80:31_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green">E40</font></i></a></i>;<br/><b>then </b><i><font color="Green">E64</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>29</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="graph_2.html#T4">GRAPH_2:4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font>

;<br/><a NAME="E82:31_2_1"/><i><font color="Green">E65</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>28</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E75:31_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E83:31_2_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E84:31_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54">FINSEQ_3:54</a></i>;<br/><a NAME="E85:31_2_1"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><a NAME="E86:31_2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E75:31_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E87:31_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E74:31_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E88:31_2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E75:31_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E89:31_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E74:31_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>set </b><font color="Maroon">c<sub>30</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>31</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_13_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_13_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_1_13_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_13_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_13_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:31_2_1_13_1"/><i><font color="Green">E73</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_13_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_1_13_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_4.html#E85:31_2_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:31_2_1_13_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E84:31_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_4.html#E85:31_2_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:31_2_1_13_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_13_1"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E10:31_2_1_13_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_13_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_13_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_1_13_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_13_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>36</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_1_13_1"/><i><font color="Green">E78</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_13_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> = 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_1_13_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_13_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>36</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E43:31_2_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_1_13_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_13_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_13_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_13_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_13_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_13_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_13_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_13"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:31_2_1_13"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_1_13"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_13"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_13"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_1_13"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E7:31_2_1_13"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_1_13"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:31_2_1_13"/><b>then </b>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E85:31_2_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:31_2_1_13"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E11:31_2_1_13"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_13"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E12:31_2_1_13"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E84:31_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_4.html#E85:31_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_13"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_13"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:31_2_1_13"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_13"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:31_2_1_13"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1_13"><i><font color="Green">E75</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>36</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:31_2_1_13"/><i><font color="Green">E79</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1_13"><i><font color="Green">E75</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E56:31_2_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_4.html#E16:31_2_1_13"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E43:31_2_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="graph_4.html#E5:31_2_1_13"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_13"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_13"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_1_13"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_4.html#E12:31_2_1_13"><i><font color="Green">E77</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:31_2_1_13"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_13"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_13"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_13"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E91:31_2_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_14_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_14_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_1_14_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_14_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_14_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:31_2_1_14_1"/><i><font color="Green">E74</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_14_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_1_14_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E81:31_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graph_4.html#E82:31_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_14_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:31_2_1_14_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E24:31_2_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E30:31_2_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_14_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="graph_2.html#T5">GRAPH_2:5</a></i>;<br/><a NAME="E9:31_2_1_14_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:31_2_1_14_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:31_2_1_14_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E89:31_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_14_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_1_14_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_14_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>36</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_1_14_1"/><i><font color="Green">E79</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_14_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> = 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_1_14_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_14_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E43:31_2_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="graph_4.html#E89:31_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_14_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_1_14_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>36</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_1_14_1"><i><font color="Green">E79</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_14_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_14_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_14_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_14_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_14_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_14"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:31_2_1_14"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_1_14"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_14"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_14"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_1_14"/><b>then </b><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E8:31_2_1_14"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E81:31_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graph_4.html#E82:31_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_14"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:31_2_1_14"/><i><font color="Green">E76</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E81:31_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graph_4.html#E82:31_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_14"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:31_2_1_14"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E24:31_2_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E30:31_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="graph_2.html#T5">GRAPH_2:5</a></i>;<br/><a NAME="E11:31_2_1_14"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_14"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>36</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_1_14"/><i><font color="Green">E78</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_1_14"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>36</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_14"><i><font color="Green">E78</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_1_14"><i><font color="Green">E78</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E43:31_2_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_14"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_14"><i><font color="Green">E77</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_1_14"><i><font color="Green">E73</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_14"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_14"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_14"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_14"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E93:31_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E94:31_2_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>29</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E39:31_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E74:31_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_4.html#E79:31_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><b>hence </b><a NAME="E95:31_2_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E87:31_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_4.html#E89:31_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_4.html#E91:31_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_4.html#E93:31_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="graph_2.html#T26">GRAPH_2:26</a></i>;<br/><b>set </b><font color="Maroon">c<sub>32</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>33</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>34</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_15_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_15_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</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/><a NAME="E4:31_2_1_15_1"/>
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  or <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span>  )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_15_1"><i><font color="Green">E72</font></i></a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_15_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_15"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_15_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_15_2"/>
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  or <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span>  )
 </span><b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_15"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_15_2_1"><b>suppose </b></a><a NAME="E1:31_2_1_15_2_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_15_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:31_2_1_15_2_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_15_2_1"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_15_2_2"><b>suppose </b></a><a NAME="E1:31_2_1_15_2_2"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_15_2_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</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/><b>thus </b><a NAME="E4:31_2_1_15_2_2"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_15_2_2"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E97:31_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E98:31_2_1"/><i><font color="Green">E73</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_16"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>34</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_2_1_16_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_16_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E4:31_2_1_16_1"/>
<font color="Maroon">c<sub>35</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="graph_4.html#E15:31_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_16_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_16_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_16_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>34</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:31_2_1_16"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:31_2_1_16"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>35</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/>
<a NAME="E5:31_2_1_16"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E6:31_2_1_16"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_16"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:31_2_1_16"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_16"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="graph_4.html#E98:31_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_17"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>36</sub></font>, <font color="Maroon">c<sub>37</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:31_2_1_17"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>34</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>38</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_17"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>38</sub></font> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>assume </b><a NAME="E4:31_2_1_17"/>
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>35</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>39</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:31_2_1_17"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>39</sub></font> &amp; <font color="Maroon">c<sub>39</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/><a NAME="E7:31_2_1_17"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:31_2_1_17"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>39</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_17"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:31_2_1_17"/>
<font color="Maroon">c<sub>38</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_17"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:31_2_1_17"/>
<font color="Maroon">c<sub>36</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_17"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_17"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_17"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E102:31_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E98:31_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/><a NAME="E103:31_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E104:31_2_1"/><b>then </b>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E105:31_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E107:31_2_1"/>
<font color="Maroon">c<sub>36</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>36</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><i><font color="Green">E75</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>37</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E18:31_2_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_4.html#E26:31_2_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1

;<br/><b>then </b><i><font color="Green">E76</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>35</sub></font> = 
<font color="Maroon">c<sub>37</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="ref" href="graph_2.html#T4">GRAPH_2:4</a></i>
<br/>.= 
<font color="Maroon">c<sub>36</sub></font>

;<br/><a NAME="E111:31_2_1"/><i><font color="Green">E77</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="graph_4.html#E111:31_2_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font>

;<br/><a NAME="E113:31_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E114:31_2_1"/><i><font color="Green">E79</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>34</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E98:31_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E115:31_2_1"/><i><font color="Green">E80</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E19:31_2_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="graph_4.html#E55:31_2_1"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E116:31_2_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E117:31_2_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54">FINSEQ_3:54</a></i>;<br/><a NAME="E118:31_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><a NAME="E119:31_2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E98:31_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E120:31_2_1"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_4.html#E97:31_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E121:31_2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E98:31_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E122:31_2_1"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_4.html#E97:31_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>set </b><font color="Maroon">c<sub>38</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>39</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_21"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>40</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_21_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>41</sub></font>, <font color="Maroon">c<sub>42</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_21_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>42</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_1_21_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_21_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_21_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>41</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:31_2_1_21_1"/><i><font color="Green">E88</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_21_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_1_21_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_4.html#E118:31_2_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:31_2_1_21_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E117:31_2_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_4.html#E118:31_2_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:31_2_1_21_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_21_1"><i><font color="Green">E88</font></i></a></i>;<br/><a NAME="E10:31_2_1_21_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_21_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_21_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_1_21_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_21_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>44</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_1_21_1"/><i><font color="Green">E93</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>44</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_21_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> = 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_1_21_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_21_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>44</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_4.html#E50:31_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_1_21_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_21_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E15:31_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_21_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_21_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_21_1"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_21_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_21_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_21"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>41</sub></font>, <font color="Maroon">c<sub>42</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:31_2_1_21"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>42</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_1_21"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_21"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_21"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_1_21"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E7:31_2_1_21"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_1_21"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:31_2_1_21"/><b>then </b>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E118:31_2_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>43</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:31_2_1_21"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E11:31_2_1_21"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_21"><i><font color="Green">E90</font></i></a></i>;<br/><a NAME="E12:31_2_1_21"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E117:31_2_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_4.html#E118:31_2_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_21"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_21"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:31_2_1_21"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_21"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:31_2_1_21"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1_21"><i><font color="Green">E90</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>44</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:31_2_1_21"/><i><font color="Green">E94</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>44</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_1_21"><i><font color="Green">E90</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E15:31_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_4.html#E115:31_2_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_4.html#E16:31_2_1_21"><i><font color="Green">E94</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E50:31_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_4.html#E5:31_2_1_21"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_21"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_21"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_1_21"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_4.html#E12:31_2_1_21"><i><font color="Green">E92</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:31_2_1_21"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_21"><i><font color="Green">E86</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_21"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_21"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E124:31_2_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E125:31_2_1"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/><a NAME="E126:31_2_1"/><i><font color="Green">E87</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_22"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>40</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_1_22_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>41</sub></font>, <font color="Maroon">c<sub>42</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_1_22_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>42</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_1_22_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_1_22_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_1_22_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>41</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:31_2_1_22_1"/><i><font color="Green">E91</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_22_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_1_22_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E110:31_2_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_4.html#E113:31_2_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_4.html#E114:31_2_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_22_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:31_2_1_22_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E109:31_2_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_4.html#E113:31_2_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_4.html#E125:31_2_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_22_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="graph_2.html#T5">GRAPH_2:5</a></i>;<br/><a NAME="E9:31_2_1_22_1"/><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:31_2_1_22_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:31_2_1_22_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E122:31_2_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_1_22_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_1_22_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_22_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>44</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_1_22_1"/><i><font color="Green">E96</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>44</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_22_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> = 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>43</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_1_22_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_22_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<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>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>43</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E50:31_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_4.html#E122:31_2_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_22_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_1_22_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>44</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_1_22_1"><i><font color="Green">E96</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E126:31_2_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_22_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_22_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_22_1"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_1_22_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_22_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_1_22"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>41</sub></font>, <font color="Maroon">c<sub>42</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:31_2_1_22"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>42</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_1_22"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_1_22"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_1_22"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_1_22"/><b>then </b><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>41</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E8:31_2_1_22"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E110:31_2_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_4.html#E113:31_2_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_4.html#E114:31_2_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_22"><i><font color="Green">E91</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:31_2_1_22"/><i><font color="Green">E93</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E110:31_2_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_4.html#E113:31_2_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_4.html#E114:31_2_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_22"><i><font color="Green">E91</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:31_2_1_22"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E109:31_2_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_4.html#E113:31_2_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_4.html#E125:31_2_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="graph_2.html#T5">GRAPH_2:5</a></i>;<br/><a NAME="E11:31_2_1_22"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_1_22"><i><font color="Green">E91</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>44</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_1_22"/><i><font color="Green">E95</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>44</sub></font> &amp; <font color="Maroon">c<sub>44</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_1_22"><i><font color="Green">E93</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>44</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E126:31_2_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_1_22"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>
<br/>.= 
<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>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>43</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_1_22"><i><font color="Green">E95</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>43</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E50:31_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_22"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_1_22"><i><font color="Green">E94</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_1_22"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_1_22"><i><font color="Green">E91</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_1_22"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_1_22"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_1_22"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E128:31_2_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E129:31_2_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>35</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_4.html#E97:31_2_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_4.html#E102:31_2_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><b>hence </b><a NAME="E130:31_2_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E54:31_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_4.html#E120:31_2_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graph_4.html#E122:31_2_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_4.html#E124:31_2_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graph_4.html#E128:31_2_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="graph_2.html#T26">GRAPH_2:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2"><b>suppose </b></a><a NAME="E1:31_2_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><a NAME="E2:31_2_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:31_2_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:31_2_2"/><i><font color="Green">E29</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:31_2_2"/><i><font color="Green">E30</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E2:31_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="graph_4.html#T13" target="_self">Th13</a></i>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E7:31_2_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self">Th14</a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:31_2_2"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E10:31_2_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E11:31_2_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_2_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</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>3</sub></font> )
 ;<br/>
<a NAME="E4:31_2_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</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>3</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E13:31_2_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_2_2"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_2_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:31_2_2_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E7:31_2_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_2_2_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:31_2_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_2"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_2_2_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E16:31_2_2"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 <b>and </b><br/><a NAME="E17:31_2_2"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</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>}</span>  holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E18:31_2_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_2_3"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</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/>
<a NAME="E4:31_2_2_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</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="graph_4.html#E4:31_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_2_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E20:31_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_2_4"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_2_4"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_2_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:31_2_2_4"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</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="graph_4.html#E16:31_2_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E7:31_2_2_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</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="graph_4.html#E4:31_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_2_2_4"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:31_2_2_4"/>
<font color="Maroon">c<sub>18</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>19</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="graph_4.html#E17:31_2_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_4"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_2_2_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/><b>take </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E22:31_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>;<br/><b>then </b><i><font color="Green">E38</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1

;<br/><a NAME="E24:31_2_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E25:31_2_2"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E26:31_2_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E28:31_2_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E29:31_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_4.html#E24:31_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E31:31_2_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/><a NAME="E32:31_2_2"/><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E33:31_2_2"/><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">c<sub>5</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>;<br/><b>then </b><i><font color="Green">E43</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1

;<br/><a NAME="E35:31_2_2"/><b>then </b><i><font color="Green">E44</font></i>: 
1 <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>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E36:31_2_2"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E37:31_2_2"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E38:31_2_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E39:31_2_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><b>hence </b><a NAME="E40:31_2_2"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E36:31_2_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/><b>thus </b><a NAME="E41:31_2_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="txt" href="graph_4.html#E5:31_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E42:31_2_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E7:31_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E43:31_2_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E40:31_2_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>thus </b><a NAME="E44:31_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>;<br/><b>thus </b><a NAME="E45:31_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>;<br/><a NAME="E46:31_2_2"/><i><font color="Green">E47</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_7"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_2_7"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<a NAME="E4:31_2_2_7"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_7"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_2_7"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_7"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E48:31_2_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_4.html#E24:31_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E50:31_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green">E38</font></i></a></i>;<br/><b>then </b><i><font color="Green">E48</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>21</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="graph_2.html#T4">GRAPH_2:4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>

;<br/><a NAME="E52:31_2_2"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E53:31_2_2"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E46:31_2_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>set </b><font color="Maroon">c<sub>23</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_2_9_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_2_9_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_2_9_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_2_9_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_9_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>25</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:31_2_2_9_1"/><i><font color="Green">E54</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_9_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_2_9_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E51:31_2_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E52:31_2_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_9_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:31_2_2_9_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_4.html#E28:31_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E31:31_2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_9_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="graph_2.html#T5">GRAPH_2:5</a></i>;<br/><a NAME="E9:31_2_2_9_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:31_2_2_9_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:31_2_2_9_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E53:31_2_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_2_9_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_2_9_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_9_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_2_9_1"/><i><font color="Green">E59</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_9_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_2_9_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_9_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="graph_4.html#E53:31_2_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_9_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_2_9_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_2_9_1"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_9_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_2_9_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:31_2_2_9_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_9_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_9_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_2_9"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:31_2_2_9"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_2_9"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_2_9"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_9"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_2_9"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>25</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E8:31_2_2_9"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E51:31_2_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E52:31_2_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_9"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:31_2_2_9"/><i><font color="Green">E56</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E51:31_2_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E52:31_2_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_9"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:31_2_2_9"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E23:31_2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_4.html#E28:31_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E31:31_2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="graph_2.html#T5">GRAPH_2:5</a></i>;<br/><a NAME="E11:31_2_2_9"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_9"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_2_9"/><i><font color="Green">E58</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2_9"><i><font color="Green">E56</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_2_9"><i><font color="Green">E58</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_2_9"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_9"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_9"><i><font color="Green">E57</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_2_9"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_9"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_2_9"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_9"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_9"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E55:31_2_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E56:31_2_2"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E57:31_2_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_2_10"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_2_10"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</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/>
<a NAME="E4:31_2_2_10"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_10"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_2_10"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_10"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E59:31_2_2"/>
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&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>5</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E35:31_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E60:31_2_2"/><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">c<sub>5</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</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>5</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <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>5</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E62:31_2_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E34:31_2_2"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E63:31_2_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</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>5</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/><i><font color="Green">E54</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>24</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E62:31_2_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="graph_2.html#T4">GRAPH_2:4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

;<br/><a NAME="E65:31_2_2"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E57:31_2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E66:31_2_2"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="graph_4.html#E57:31_2_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>set </b><font color="Maroon">c<sub>26</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_2_12_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_2_12_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_2_12_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_2_12_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_2_12_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>28</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:31_2_2_12_1"/><i><font color="Green">E60</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>30</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>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_12_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_2_12_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E64:31_2_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_4.html#E65:31_2_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_12_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:31_2_2_12_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E28:31_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E34:31_2_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="graph_4.html#E63:31_2_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_12_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="graph_2.html#T5">GRAPH_2:5</a></i>;<br/><a NAME="E9:31_2_2_12_1"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E7:31_2_2_12_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:31_2_2_12_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E66:31_2_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_2_12_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_2_12_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_12_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_2_12_1"/><i><font color="Green">E65</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_12_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_2_12_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_12_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<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>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="graph_4.html#E66:31_2_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_12_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_2_12_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<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>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_2_12_1"><i><font color="Green">E65</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E32:31_2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_12_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_2_12_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:31_2_2_12_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_2_12_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_12_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_2_12"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:31_2_2_12"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_2_12"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_2_12"><i><font color="Green">E57</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_2_12"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_2_12"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>28</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E8:31_2_2_12"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E64:31_2_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_4.html#E65:31_2_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_12"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:31_2_2_12"/><i><font color="Green">E62</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>30</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>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E64:31_2_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graph_4.html#E65:31_2_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_12"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:31_2_2_12"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="graph_4.html#E28:31_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E34:31_2_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="graph_4.html#E63:31_2_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="graph_2.html#T5">GRAPH_2:5</a></i>;<br/><a NAME="E11:31_2_2_12"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_2_12"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_2_12"/><i><font color="Green">E64</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_2_12"><i><font color="Green">E62</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E32:31_2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_2_12"><i><font color="Green">E64</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>
<br/>.= 
<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>19</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_2_12"><i><font color="Green">E64</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_12"><i><font color="Green">E60</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_2_12"><i><font color="Green">E63</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_2_12"><i><font color="Green">E59</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_2_12"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_2_12"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_2_12"><i><font color="Green">E58</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_2_12"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E68:31_2_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E69:31_2_2"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E16:31_2_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_3"><b>suppose </b></a><a NAME="E1:31_2_3"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><a NAME="E2:31_2_3"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:31_2_3"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:31_2_3"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:31_2_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:31_2_3"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:31_2_3"/><i><font color="Green">E29</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = 1,<font color="Maroon">c<sub>6</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font> as  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="graph_4.html#T13" target="_self">Th13</a></i>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> = 1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E9:31_2_3"/><i><font color="Green">E30</font></i>: 
( 1 <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> )
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:31_2_3"/><i><font color="Green">E31</font></i>: 
1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="txt" href="graph_4.html#E9:31"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E11:31_2_3"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="graph_4.html#D5" target="_self">Def5</a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E13:31_2_3"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E14:31_2_3"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E15:31_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_3_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</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> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>
<a NAME="E4:31_2_3_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</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>3</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E17:31_2_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_3_2"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_3_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_3_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:31_2_3_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E7:31_2_3_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_2_3_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_3_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:31_2_3_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_4.html#E14:31_2_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_2"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_2_3_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_3_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E20:31_2_3"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E21:31_2_3"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E22:31_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_3_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<a NAME="E4:31_2_3_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</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="graph_4.html#E9:31_2_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_3_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E24:31_2_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_3_4"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:31_2_3_4"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_3_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_4"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:31_2_3_4"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E7:31_2_3_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</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="graph_4.html#E9:31_2_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_2_3_4"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_3_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:31_2_3_4"/>
<font color="Maroon">c<sub>18</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>19</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_4.html#E21:31_2_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_4"><i><font color="Green">E41</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_4"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:31_2_3_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_4"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_3_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/><b>take </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>18</sub></font> = 1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E26:31_2_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>;<br/><b>then </b><i><font color="Green">E39</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="graph_4.html#E7:31"><i><font color="Green">E23</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><b>hence </b><a NAME="E28:31_2_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green">E28</font></i></a></i>;<br/><b>thus </b><a NAME="E29:31_2_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="txt" href="graph_4.html#E9:31"><i><font color="Green">E25</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="graph_4.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E30:31_2_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="graph_4.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E31:31_2_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&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="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E32:31_2_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>;<br/><b>thus </b><a NAME="E33:31_2_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="graph_4.html#E1:31_2_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>;<br/><a NAME="E34:31_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_6"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_3_6"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>
<a NAME="E4:31_2_3_6"/>
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_6"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_3_6"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_6"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E36:31_2_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E37:31_2_3"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54">FINSEQ_3:54</a></i>;<br/><a NAME="E38:31_2_3"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_3_7_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_3_7_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_3_7_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_3_7_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_7_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:31_2_3_7_1"/><i><font color="Green">E45</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_7_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_3_7_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E38:31_2_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:31_2_3_7_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E37:31_2_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E38:31_2_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:31_2_3_7_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_7_1"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E10:31_2_3_7_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3_7_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_3_7_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_3_7_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_7_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_3_7_1"/><i><font color="Green">E50</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_7_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_3_7_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_7_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_4.html#E17:31_2_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_3_7_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_7_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_7_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_7_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:31_2_3_7_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_7_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_7_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_3_7"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:31_2_3_7"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_3_7"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_3_7"><i><font color="Green">E42</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_7"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_3_7"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E7:31_2_3_7"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_3_7"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:31_2_3_7"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E38:31_2_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:31_2_3_7"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E11:31_2_3_7"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_7"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E12:31_2_3_7"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E37:31_2_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="graph_4.html#E38:31_2_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3_7"><i><font color="Green">E46</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_7"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:31_2_3_7"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_7"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:31_2_3_7"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_3_7"><i><font color="Green">E47</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:31_2_3_7"/><i><font color="Green">E51</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_3_7"><i><font color="Green">E47</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E16:31_2_3_7"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graph_4.html#E30"><i><font color="Green">Lemma17</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E17:31_2_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="graph_4.html#E5:31_2_3_7"><i><font color="Green">E44</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_7"><i><font color="Green">E45</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_7"><i><font color="Green">E47</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_3_7"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E12:31_2_3_7"><i><font color="Green">E49</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:31_2_3_7"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_7"><i><font color="Green">E43</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_7"><i><font color="Green">E47</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_7"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E40:31_2_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E41:31_2_3"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E13:31_2_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E42:31_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2_3_8"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_3_8"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<a NAME="E4:31_2_3_8"/>
<font color="Maroon">c<sub>22</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="graph_4.html#E9:31_2_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_8"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:31_2_3_8"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_8"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E44:31_2_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E45:31_2_3"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54">FINSEQ_3:54</a></i>;<br/><a NAME="E46:31_2_3"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_2_3_9_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> 1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_2_3_9_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:31_2_3_9_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E1:31_2_3_9_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="graph_4.html#E3:31_2_3_9_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:31_2_3_9_1"/><i><font color="Green">E47</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_9_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:31_2_3_9_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E46:31_2_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:31_2_3_9_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E45:31_2_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="graph_4.html#E46:31_2_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:31_2_3_9_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_9_1"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E10:31_2_3_9_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3_9_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E8:31_2_3_9_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:31_2_3_9_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_9_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:31_2_3_9_1"/><i><font color="Green">E52</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_9_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E8:31_2_3_9_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_9_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_4.html#E24:31_2_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_4.html#E9:31_2_3_9_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_9_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_9_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_9_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
;<br/><b>hence </b><a NAME="E15:31_2_3_9_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E3:31_2_3_9_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_9_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_2_3_9"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:31_2_3_9"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:31_2_3_9"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E2:31_2_3_9"><i><font color="Green">E44</font></i></a>, <a class="txt" href="graph_4.html#E4:31_2_3_9"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:31_2_3_9"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E7:31_2_3_9"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="graph_4.html#E5:31_2_3_9"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:31_2_3_9"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="graph_4.html#E46:31_2_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:31_2_3_9"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E11:31_2_3_9"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_9"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E12:31_2_3_9"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E45:31_2_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="graph_4.html#E46:31_2_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="graph_4.html#E7:31_2_3_9"><i><font color="Green">E48</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_9"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:31_2_3_9"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default">1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_9"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:31_2_3_9"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_3_9"><i><font color="Green">E49</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:31_2_3_9"/><i><font color="Green">E53</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_4.html#E10:31_2_3_9"><i><font color="Green">E49</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><span class="p1">(<span class="default">1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E9:31_2_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="graph_4.html#E27:31_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="graph_4.html#E16:31_2_3_9"><i><font color="Green">E53</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="graph_4.html#E24:31_2_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="graph_4.html#E5:31_2_3_9"><i><font color="Green">E46</font></i></a>, <a class="txt" href="graph_4.html#E6:31_2_3_9"><i><font color="Green">E47</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_9"><i><font color="Green">E49</font></i></a>, <a class="txt" href="graph_4.html#E11:31_2_3_9"><i><font color="Green">E50</font></i></a>, <a class="txt" href="graph_4.html#E12:31_2_3_9"><i><font color="Green">E51</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:31_2_3_9"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> 1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E4:31_2_3_9"><i><font color="Green">E45</font></i></a>, <a class="txt" href="graph_4.html#E10:31_2_3_9"><i><font color="Green">E49</font></i></a>, <a class="txt" href="graph_4.html#E13:31_2_3_9"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E48:31_2_3"/><b>then </b>
1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E49:31_2_3"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="graph_4.html#E20:31_2_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
