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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#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:71"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is  not  <a href="graph_2.html#V3" target="_self">simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:71"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>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="E4:71"/><i><font color="Green">E62</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="txt" href="graph_2.html#E1:71"><i><font color="Green">E60</font></i></a>, <a class="txt" href="graph_2.html#E2:71"><i><font color="Green">E61</font></i></a>, <a class="ref" href="graph_2.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E5:71"/><i><font color="Green">E63</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_2.html#E2:71"><i><font color="Green">E61</font></i></a>, <a class="ref" href="graph_2.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E6:71"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</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="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="graph_2.html#E4:71"><i><font color="Green">E62</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="E8:71"/><i><font color="Green">E65</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="graph_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E9:71"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</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/>
<a NAME="E10:71"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1"/>
( ( <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_2.html#E4:71"><i><font color="Green">E62</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1"><b>suppose </b></a><a NAME="E1:71_1_1"/><i><font color="Green">E68</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="E2:71_1_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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:71_1_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="E4:71_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:71_1_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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:71_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</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="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:71_1_1"/><i><font color="Green">E71</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="E8:71_1_1"/><i><font color="Green">E72</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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E1:71_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:71_1_1"/><b>then </b><i><font color="Green">E73</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_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:71_1_1"/><i><font color="Green">E74</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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_1"><i><font color="Green">E72</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" target="_self">-cut</a> <font color="Maroon">c<sub>3</sub></font> as    <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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="graph_2.html#T44" target="_self">Th44</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" target="_self">-cut</a> <font color="Maroon">c<sub>3</sub></font> as    <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_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="graph_2.html#T44" target="_self">Th44</a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = 1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2" target="_self">-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>11</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" target="_self">-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="E15:71_1_1"/><i><font color="Green">E75</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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:71_1_1"/><i><font color="Green">E76</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_2.html#E2:71"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="graph_2.html#D7" target="_self">Def7</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:71_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E2:71"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_2.html#E10:71"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="graph_2.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E18:71_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E2:71"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="graph_2.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E19:71_1_1"/><i><font color="Green">E79</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</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_2.html#E17:71_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="graph_2.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E20:71_1_1"/><i><font color="Green">E80</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</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_2.html#E18:71_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="graph_2.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E21:71_1_1"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E22:71_1_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="xcmplx_0.html#K6">-</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>12</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E24:71_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E25:71_1_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_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="graph_2.html#D1" target="_self">Def1</a></i>;<br/><b>then </b><i><font color="Green">E82</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="xcmplx_0.html#K6">-</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="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1

;<br/><a NAME="E27:71_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</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="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E9:71_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E28:71_1_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="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E82</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>13</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="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E30:71_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</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="E31:71_1_1"/><i><font color="Green">E85</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>11</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E20:71_1_1"><i><font color="Green">E80</font></i></a></i>;<br/><a NAME="E32:71_1_1"/><b>then </b><i><font color="Green">E86</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>11</sub></font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E87</font></i>: <font color="Maroon">c<sub>11</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>11</sub></font></span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="graph_2.html#T7" target="_self">Th7</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" target="_self">-cut</a> <font color="Maroon">c<sub>11</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>11</sub></font></span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E32:71_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="graph_2.html#T8" target="_self">Th8</a></i>
;<br/><a NAME="E34:71_1_1"/>
<font color="Maroon">c<sub>12</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_2.html#E24:71_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E35:71_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>11</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>12</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" target="_self">-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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="ref" href="graph_2.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E36:71_1_1"/><i><font color="Green">E89</font></i>: 
<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
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>1,1 <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">c<sub>11</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>11</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_2.html#E32:71_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</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_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E31:71_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graph_2.html#E36:71_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</a></i>
;<br/><a NAME="E38:71_1_1"/><b>then </b><i><font color="Green">E90</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" target="_self">-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>11</sub></font></span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E33:71_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_2.html#E35:71_1_1"><i><font color="Green">E88</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>14</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>15</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E40:71_1_1"/><i><font color="Green">E91</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</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="E41:71_1_1"/><i><font color="Green">E92</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>15</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="E42:71_1_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="71_1_1_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="E1:71_1_1_4"/>
<font color="Maroon">c<sub>16</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>17</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_4"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</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>17</sub></font> &amp; <font color="Maroon">c<sub>17</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="E4:71_1_1_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_1_4"><i><font color="Green">E93</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_1_4"/>
<font color="Maroon">c<sub>16</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_2.html#E3:71_1_1_4"><i><font color="Green">E93</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>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_2.html#E40:71_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E44:71_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>16</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="71_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</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="E1:71_1_1_5"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_5"/><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:71_1_1_5"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>18</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> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_2.html#E1:71_1_1_5"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_1_5"><i><font color="Green">E95</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>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:71_1_1_5"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>18</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> ) or ( <font color="Maroon">c<sub>5</sub></font> <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>  <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_2.html#E40:71_1_1"><i><font color="Green">E91</font></i></a></i>;<br/>
<a NAME="E7:71_1_1_5"/>
( 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>  <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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_5"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:71_1_1_5"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>18</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_2.html#E6:71_1_1_5"><i><font color="Green">E97</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:71_1_1_5"/>
<font color="Maroon">c<sub>19</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>20</sub></font>
 
<b>by </b><i><a class="txt" href="graph_2.html#E40:71_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E41:71_1_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_5"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_5"><i><font color="Green">E97</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:71_1_1_5"/>
<font color="Maroon">c<sub>17</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_2.html#E3:71_1_1_5"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_5"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_1_5"><i><font color="Green">E98</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>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="graph_2.html#M1" target="_self">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5" target="_self">Def5</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/><b>set </b><font color="Maroon">c<sub>18</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>19</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E47:71_1_1"/><i><font color="Green">E94</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</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="E48:71_1_1"/><i><font color="Green">E95</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>19</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="E49:71_1_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="71_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</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="E1:71_1_1_6"/>
<font color="Maroon">c<sub>20</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>21</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_6"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</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>21</sub></font> &amp; <font color="Maroon">c<sub>21</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="E4:71_1_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="E5:71_1_1_6"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</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_2.html#E15:71_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_1_6"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:71_1_1_6"/>
<font color="Maroon">c<sub>20</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_2.html#E3:71_1_1_6"><i><font color="Green">E96</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>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="txt" href="graph_2.html#E47:71_1_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E51:71_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>20</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="71_1_1_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="E1:71_1_1_7"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</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="E3:71_1_1_7"/><i><font color="Green">E98</font></i>: 
<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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:71_1_1_7"/><i><font color="Green">E99</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> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</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_2.html#E1:71_1_1_7"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_1_7"><i><font color="Green">E98</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>24</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:71_1_1_7"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">c<sub>22</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>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>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>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>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="graph_2.html#E47:71_1_1"><i><font color="Green">E94</font></i></a></i>;<br/>
<a NAME="E7:71_1_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="E8:71_1_1_7"/><b>then </b>
( 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>2</sub></font> )
 
<b>by </b><i><a class="txt" href="graph_2.html#E15:71_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_7"><i><font color="Green">E100</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:71_1_1_7"/><b>then </b><i><font color="Green">E101</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>2</sub></font>
 
<b>by </b><i><a class="txt" href="graph_2.html#E6:71_1_1_7"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:71_1_1_7"/>
<font color="Maroon">c<sub>23</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>24</sub></font>
 
<b>by </b><i><a class="txt" href="graph_2.html#E47:71_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_2.html#E48:71_1_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_7"><i><font color="Green">E99</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_7"><i><font color="Green">E100</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:71_1_1_7"/>
<font color="Maroon">c<sub>21</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_2.html#E3:71_1_1_7"><i><font color="Green">E98</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_7"><i><font color="Green">E100</font></i></a>, <a class="txt" href="graph_2.html#E9:71_1_1_7"><i><font color="Green">E101</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>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <a href="graph_2.html#M1" target="_self">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5" target="_self">Def5</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>21</sub></font>
;<br/><a NAME="E53:71_1_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>11</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>10</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>10</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_2.html#E15:71_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_2.html#E16:71_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_2.html#T12" target="_self">Th12</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</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#M2">Chain</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="graph_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E17:71_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_2.html#E18:71_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="graph_2.html#T46" target="_self">Th46</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>22</sub></font>
;<br/><b>take </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="graph_2.html#K4" target="_self">^'</a> <font color="Maroon">c<sub>11</sub></font>;<br/><a NAME="E55:71_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E38:71_1_1"><i><font color="Green">E90</font></i></a></i>;<br/><a NAME="E56:71_1_1"/><i><font color="Green">E99</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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/><a NAME="E57:71_1_1"/><b>then </b><i><font color="Green">E100</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="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E9:71"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E58:71_1_1"/>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</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="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E59:71_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</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_2.html#E6:71"><i><font color="Green">E64</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>22</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</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="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E57:71_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</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="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E61:71_1_1"/><i><font color="Green">E102</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>22</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_2.html#E59:71_1_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/><b>thus </b><a NAME="E62:71_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E17:71_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_2.html#E18:71_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_2.html#E53:71_1_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="graph_2.html#T47" target="_self">Th47</a></i>;<br/><a NAME="E63:71_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</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>22</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E64:71_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</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_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E61:71_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E65:71_1_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="E66:71_1_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>10</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E17:71_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="graph_2.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E67:71_1_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E68:71_1_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>23</sub></font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E15:71_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="graph_2.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E69:71_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>11</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>11</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_2.html#E16:71_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="graph_2.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E70:71_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</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="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E9:71_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E71:71_1_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="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E72:71_1_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>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E20:71_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E73:71_1_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>23</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>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E69:71_1_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="graph_2.html#T16" target="_self">Th16</a></i>;<br/><b>set </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> : ( 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>25</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="71_1_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_9_1"/>
<font color="Maroon">c<sub>26</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>27</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_9_1"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; ( ( 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> <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>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>3</sub></font> ) ) )
 ;<br/><a NAME="E4:71_1_1_9_1"/>
( <font color="Maroon">c<sub>26</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>26</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_2.html#E3:71_1_1_9_1"><i><font color="Green">E104</font></i></a></i>;<br/><b>hence </b><a NAME="E5:71_1_1_9_1"/>
<font color="Maroon">c<sub>26</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="E2:71_1_1_9"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>26</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="71_1_1_9_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_1_9_2"/>
( <font color="Maroon">c<sub>26</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>26</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_2.html#E2:71_1_1_9"><i><font color="Green">E104</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="71_1_1_9_2_1"><b>suppose </b></a><a NAME="E1:71_1_1_9_2_1"/>
<font color="Maroon">c<sub>26</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>27</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_9_2_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; 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> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:71_1_1_9_2_1"/>
<font color="Maroon">c<sub>26</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_2.html#E3:71_1_1_9_2_1"><i><font color="Green">E105</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_9_2_2"><b>suppose </b></a><a NAME="E1:71_1_1_9_2_2"/>
<font color="Maroon">c<sub>26</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>27</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_9_2_2"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <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>3</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:71_1_1_9_2_2"/>
<font color="Maroon">c<sub>26</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_2.html#E3:71_1_1_9_2_2"><i><font color="Green">E105</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E75:71_1_1"/><b>then </b><i><font color="Green">E104</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="E76:71_1_1"/><i><font color="Green">E105</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="71_1_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>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:71_1_1_10_1"/>
<font color="Maroon">c<sub>26</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>27</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_10_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; 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> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E4:71_1_1_10_1"/>
<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>3</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_1_10_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:71_1_1_10_1"/>
<font color="Maroon">c<sub>26</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_2.html#E3:71_1_1_10_1"><i><font color="Green">E106</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>26</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="E2:71_1_1_10"/>
<font color="Maroon">c<sub>26</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>27</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:71_1_1_10"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <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>3</sub></font> )
 ;<br/>
<a NAME="E5:71_1_1_10"/>
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_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_10"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:71_1_1_10"/>
<font color="Maroon">c<sub>26</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_2.html#E4:71_1_1_10"><i><font color="Green">E106</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>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>  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>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>  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_2.html#E76:71_1_1"><i><font color="Green">E105</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="71_1_1_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:71_1_1_11"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>30</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_11"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> &amp; 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> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><b>assume </b><a NAME="E4:71_1_1_11"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font>
 ;<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="E6:71_1_1_11"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">c<sub>31</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>31</sub></font> &amp; <font color="Maroon">c<sub>31</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="E7:71_1_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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E10:71"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:71_1_1_11"/><b>then </b>
<font color="Maroon">c<sub>30</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_2.html#E3:71_1_1_11"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E9:71_1_1_11"/>
<font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E3:71_1_1_11"><i><font color="Green">E106</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_11"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E80:71_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>27</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>26</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>27</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E76:71_1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/><a NAME="E81:71_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>13</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_2.html#E26:71_1_1"><i><font color="Green">E82</font></i></a></i>;<br/><b>then </b><i><font color="Green">E107</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>27</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="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font>

;<br/><a NAME="E83:71_1_1"/><i><font color="Green">E108</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>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>26</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>27</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E76:71_1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E84:71_1_1"/>
<font color="Maroon">c<sub>26</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="E85:71_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>26</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="E86:71_1_1"/><b>then </b><i><font color="Green">E110</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>26</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="E87:71_1_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>26</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E76:71_1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E88:71_1_1"/><b>then </b><i><font color="Green">E111</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>26</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E40:71_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E75:71_1_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E89:71_1_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>27</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E76:71_1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E90:71_1_1"/><b>then </b><i><font color="Green">E112</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>27</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E40:71_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E75:71_1_1"><i><font color="Green">E104</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>28</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>26</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>29</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>27</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_13_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>30</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>31</sub></font>, <font color="Maroon">c<sub>32</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_13_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>30</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>31</sub></font>,<font color="Maroon">c<sub>32</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="E4:71_1_1_13_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>31</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>32</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>31</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E1:71_1_1_13_1"><i><font color="Green">E113</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_1_13_1"><i><font color="Green">E114</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>33</sub></font> = <font color="Maroon">c<sub>31</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:71_1_1_13_1"/><i><font color="Green">E116</font></i>: 
( 1 <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>7</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_1_13_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_1_13_1"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>31</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>26</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E57:71_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="graph_2.html#E86:71_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:71_1_1_13_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>33</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>26</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E85:71_1_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="graph_2.html#E86:71_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:71_1_1_13_1"/><i><font color="Green">E119</font></i>: 
<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> ) 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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E57:71_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_13_1"><i><font color="Green">E116</font></i></a></i>;<br/><a NAME="E10:71_1_1_13_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>31</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>17</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>26</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E40:71_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_1_13_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_1_13_1"><i><font color="Green">E118</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_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>33</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_1_13_1"><i><font color="Green">E115</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>34</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:71_1_1_13_1"/><i><font color="Green">E121</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>34</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>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E57:71_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_13_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> = 
<font color="Maroon">c<sub>17</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_2.html#E8:71_1_1_13_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_13_1"><i><font color="Green">E120</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>34</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E40:71_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E44:71_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_2.html#E9:71_1_1_13_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_1_13_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E57:71_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_13_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_1_13_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:71_1_1_13_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</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>26</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E3:71_1_1_13_1"><i><font color="Green">E114</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_13_1"><i><font color="Green">E120</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="E2:71_1_1_13"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</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>26</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>31</sub></font>, <font color="Maroon">c<sub>32</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_1_13"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>30</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>31</sub></font>,<font color="Maroon">c<sub>32</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="E5:71_1_1_13"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>31</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>17</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>26</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E2:71_1_1_13"><i><font color="Green">E113</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_13"><i><font color="Green">E114</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_1_13"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>26</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</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>26</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E7:71_1_1_13"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>31</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>26</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E5:71_1_1_13"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:71_1_1_13"/><b>then </b>
<font color="Maroon">c<sub>31</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_2.html#E86:71_1_1"><i><font color="Green">E110</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>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:71_1_1_13"/><i><font color="Green">E118</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; 1 <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> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E11:71_1_1_13"/><i><font color="Green">E119</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> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E40:71_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_13"><i><font color="Green">E118</font></i></a></i>;<br/><a NAME="E12:71_1_1_13"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>33</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>26</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E85:71_1_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="graph_2.html#E86:71_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_1_13"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_13"><i><font color="Green">E118</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:71_1_1_13"/><i><font color="Green">E121</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>
 <b>by </b><i><a class="txt" href="graph_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E57:71_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_13"><i><font color="Green">E118</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:71_1_1_13"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E10:71_1_1_13"><i><font color="Green">E118</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>34</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:71_1_1_13"/><i><font color="Green">E122</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>34</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>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_2.html#E10:71_1_1_13"><i><font color="Green">E118</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</a></i>;<br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> = 
<font color="Maroon">c<sub>3</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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E57:71_1_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="graph_2.html#E16:71_1_1_13"><i><font color="Green">E122</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E44:71_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_2.html#E5:71_1_1_13"><i><font color="Green">E115</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_13"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_13"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graph_2.html#E11:71_1_1_13"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graph_2.html#E12:71_1_1_13"><i><font color="Green">E120</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:71_1_1_13"/>
<font color="Maroon">c<sub>30</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_2.html#E4:71_1_1_13"><i><font color="Green">E114</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_13"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_1_13"><i><font color="Green">E121</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E92:71_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</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>26</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="71_1_1_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_14_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>30</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>31</sub></font>, <font color="Maroon">c<sub>32</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_14_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>30</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>31</sub></font>,<font color="Maroon">c<sub>32</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="E4:71_1_1_14_1"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>31</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>32</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>31</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E1:71_1_1_14_1"><i><font color="Green">E114</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_1_14_1"><i><font color="Green">E115</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>33</sub></font> = <font color="Maroon">c<sub>31</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:71_1_1_14_1"/><i><font color="Green">E117</font></i>: 
( 1 <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>8</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_1_14_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_1_14_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>31</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>27</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E82:71_1_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="graph_2.html#E83:71_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_14_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:71_1_1_14_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</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>27</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E30:71_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_14_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="graph_2.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E9:71_1_1_14_1"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>33</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>27</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E7:71_1_1_14_1"><i><font color="Green">E118</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:71_1_1_14_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>31</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>17</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>27</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E90:71_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_1_14_1"><i><font color="Green">E118</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_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>33</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_1_14_1"><i><font color="Green">E116</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>34</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:71_1_1_14_1"/><i><font color="Green">E122</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>34</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>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_2.html#E6:71_1_1_14_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>27</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> = 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E8:71_1_1_14_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_14_1"><i><font color="Green">E121</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>12</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E44:71_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_2.html#E90:71_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_1_14_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graph_2.html#E9:71_1_1_14_1"><i><font color="Green">E120</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>34</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E13:71_1_1_14_1"><i><font color="Green">E122</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_14_1"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_1_14_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="graph_2.html#D1" target="_self">Def1</a></i>
;<br/><b>hence </b><a NAME="E15:71_1_1_14_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</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>27</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E3:71_1_1_14_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_14_1"><i><font color="Green">E121</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="E2:71_1_1_14"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</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>27</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>31</sub></font>, <font color="Maroon">c<sub>32</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_1_14"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>30</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>31</sub></font>,<font color="Maroon">c<sub>32</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="E5:71_1_1_14"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>31</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>17</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>27</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</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>27</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E2:71_1_1_14"><i><font color="Green">E114</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_14"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_1_14"/><b>then </b><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>31</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>27</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</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>33</sub></font> = <font color="Maroon">c<sub>31</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E8:71_1_1_14"/><i><font color="Green">E118</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>
 <b>by </b><i><a class="txt" href="graph_2.html#E82:71_1_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="graph_2.html#E83:71_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_14"><i><font color="Green">E117</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:71_1_1_14"/><i><font color="Green">E119</font></i>: 
( 1 <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>8</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E82:71_1_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="graph_2.html#E83:71_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_14"><i><font color="Green">E117</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:71_1_1_14"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</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>27</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E30:71_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="graph_2.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E11:71_1_1_14"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E6:71_1_1_14"><i><font color="Green">E117</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>34</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:71_1_1_14"/><i><font color="Green">E121</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>34</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>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_2.html#E9:71_1_1_14"><i><font color="Green">E119</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</a></i>;<br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_1_14"><i><font color="Green">E121</font></i></a>, <a class="ref" href="graph_2.html#D1" target="_self">Def1</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>12</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E13:71_1_1_14"><i><font color="Green">E121</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>17</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>27</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E44:71_1_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_14"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_14"><i><font color="Green">E120</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>32</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E5:71_1_1_14"><i><font color="Green">E116</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_14"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:71_1_1_14"/>
<font color="Maroon">c<sub>30</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_2.html#E4:71_1_1_14"><i><font color="Green">E115</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_1_14"><i><font color="Green">E118</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E94:71_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</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>27</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E95:71_1_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</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>26</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>27</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E40:71_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E75:71_1_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="graph_2.html#E80:71_1_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><b>hence </b><a NAME="E96:71_1_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E88:71_1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="graph_2.html#E90:71_1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="graph_2.html#E92:71_1_1"><i><font color="Green">E113</font></i></a>, <a class="txt" href="graph_2.html#E94:71_1_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="graph_2.html#T26" target="_self">Th26</a></i>;<br/><b>set </b><font color="Maroon">c<sub>30</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>31</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="71_1_1_15"><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="E1:71_1_1_15_1"/>
<font color="Maroon">c<sub>32</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>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_15_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>32</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>33</sub></font> &amp; <font color="Maroon">c<sub>33</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>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>2</sub></font> ) ) )
 ;<br/><a NAME="E4:71_1_1_15_1"/>
( <font color="Maroon">c<sub>32</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>32</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_2.html#E3:71_1_1_15_1"><i><font color="Green">E115</font></i></a></i>;<br/><b>hence </b><a NAME="E5:71_1_1_15_1"/>
<font color="Maroon">c<sub>32</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="E2:71_1_1_15"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>32</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="71_1_1_15_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_1_15_2"/>
( <font color="Maroon">c<sub>32</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>32</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_2.html#E2:71_1_1_15"><i><font color="Green">E115</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="71_1_1_15_2_1"><b>suppose </b></a><a NAME="E1:71_1_1_15_2_1"/>
<font color="Maroon">c<sub>32</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>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_15_2_1"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>32</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>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:71_1_1_15_2_1"/>
<font color="Maroon">c<sub>32</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_2.html#E3:71_1_1_15_2_1"><i><font color="Green">E116</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_15_2_2"><b>suppose </b></a><a NAME="E1:71_1_1_15_2_2"/>
<font color="Maroon">c<sub>32</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>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_15_2_2"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</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>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>2</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:71_1_1_15_2_2"/>
<font color="Maroon">c<sub>32</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_2.html#E3:71_1_1_15_2_2"><i><font color="Green">E116</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E98:71_1_1"/><b>then </b><i><font color="Green">E115</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="E99:71_1_1"/><i><font color="Green">E116</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="71_1_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>32</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:71_1_1_16_1"/>
<font color="Maroon">c<sub>32</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>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_16_1"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>32</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>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E4:71_1_1_16_1"/>
<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>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E15:71_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_1_16_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:71_1_1_16_1"/>
<font color="Maroon">c<sub>32</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_2.html#E3:71_1_1_16_1"><i><font color="Green">E117</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>32</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="E2:71_1_1_16"/>
<font color="Maroon">c<sub>32</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>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:71_1_1_16"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</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>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>2</sub></font> )
 ;<br/>
<a NAME="E5:71_1_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="E6:71_1_1_16"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_1_16"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:71_1_1_16"/>
<font color="Maroon">c<sub>32</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_2.html#E4:71_1_1_16"><i><font color="Green">E117</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>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>  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>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>  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_2.html#E99:71_1_1"><i><font color="Green">E116</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="71_1_1_17"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:71_1_1_17"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font>
 ;<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="E3:71_1_1_17"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> &amp; 1 <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#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>assume </b><a NAME="E4:71_1_1_17"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>33</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>37</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:71_1_1_17"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">c<sub>37</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>37</sub></font> &amp; <font color="Maroon">c<sub>37</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="E7:71_1_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="E8:71_1_1_17"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E6:71_1_1_17"><i><font color="Green">E118</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:71_1_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>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_1_17"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:71_1_1_17"/>
<font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E3:71_1_1_17"><i><font color="Green">E117</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_17"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_1_17"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E103:71_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>33</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>32</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>33</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E99:71_1_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/><a NAME="E104:71_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E20:71_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E105:71_1_1"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</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>11</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E106:71_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</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>11</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</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>34</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E108:71_1_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E20:71_1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E27:71_1_1"><i><font color="Green">E83</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>35</sub></font> = <font color="Maroon">c<sub>34</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><i><font color="Green">E118</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>35</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</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="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E18:71_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="graph_2.html#D7" target="_self">Def7</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">E119</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>33</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="ref" href="graph_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
<font color="Maroon">c<sub>34</sub></font>

;<br/><a NAME="E112:71_1_1"/><i><font color="Green">E120</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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</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>11</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_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E36:71_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></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" target="_self">-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_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E112:71_1_1"><i><font color="Green">E120</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</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" target="_self">-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="E114:71_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</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" target="_self">-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="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E115:71_1_1"/><i><font color="Green">E122</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>32</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>32</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>33</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E99:71_1_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E116:71_1_1"/><i><font color="Green">E123</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E19:71_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_2.html#E56:71_1_1"><i><font color="Green">E99</font></i></a></i>;<br/><a NAME="E117:71_1_1"/>
<font color="Maroon">c<sub>32</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="E118:71_1_1"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>32</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="E119:71_1_1"/><b>then </b><i><font color="Green">E125</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>32</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="E120:71_1_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>32</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E99:71_1_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E121:71_1_1"/><b>then </b><i><font color="Green">E126</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>32</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E47:71_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_2.html#E98:71_1_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E122:71_1_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>33</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E99:71_1_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E123:71_1_1"/><b>then </b><i><font color="Green">E127</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>33</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E47:71_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_2.html#E98:71_1_1"><i><font color="Green">E115</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>36</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>32</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>37</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>33</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_21"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>38</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_21_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>39</sub></font>, <font color="Maroon">c<sub>40</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_21_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>38</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>39</sub></font>,<font color="Maroon">c<sub>40</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="E4:71_1_1_21_1"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E1:71_1_1_21_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_1_21_1"><i><font color="Green">E129</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>41</sub></font> = <font color="Maroon">c<sub>39</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:71_1_1_21_1"/><i><font color="Green">E131</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_1_21_1"><i><font color="Green">E130</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_1_21_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>39</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>32</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E116:71_1_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="graph_2.html#E119:71_1_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:71_1_1_21_1"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>41</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>32</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E118:71_1_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="graph_2.html#E119:71_1_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:71_1_1_21_1"/><i><font color="Green">E134</font></i>: 
<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> ) 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_2.html#E116:71_1_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_21_1"><i><font color="Green">E131</font></i></a></i>;<br/><a NAME="E10:71_1_1_21_1"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>39</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>21</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>32</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E47:71_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_1_21_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_1_21_1"><i><font color="Green">E133</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_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>41</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_1_21_1"><i><font color="Green">E130</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>42</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:71_1_1_21_1"/><i><font color="Green">E136</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>42</sub></font> &amp; <font color="Maroon">c<sub>42</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>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_2.html#E116:71_1_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_21_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>21</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>32</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font> = 
<font color="Maroon">c<sub>21</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_2.html#E8:71_1_1_21_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_21_1"><i><font color="Green">E135</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>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E47:71_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_2.html#E51:71_1_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_2.html#E9:71_1_1_21_1"><i><font color="Green">E134</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_1_21_1"><i><font color="Green">E136</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>40</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E15:71_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_2.html#E116:71_1_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_21_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_1_21_1"><i><font color="Green">E136</font></i></a>, <a class="ref" href="graph_2.html#D1" target="_self">Def1</a></i>
;<br/><b>hence </b><a NAME="E15:71_1_1_21_1"/>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</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>32</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E3:71_1_1_21_1"><i><font color="Green">E129</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_21_1"><i><font color="Green">E135</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="E2:71_1_1_21"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</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>32</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>39</sub></font>, <font color="Maroon">c<sub>40</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_1_21"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>38</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>39</sub></font>,<font color="Maroon">c<sub>40</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="E5:71_1_1_21"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">c<sub>39</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>21</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>32</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</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>32</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E2:71_1_1_21"><i><font color="Green">E128</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_21"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_1_21"/><b>then </b><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</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>32</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</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>32</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E7:71_1_1_21"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">c<sub>39</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>32</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>32</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E5:71_1_1_21"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:71_1_1_21"/><b>then </b>
<font color="Maroon">c<sub>39</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_2.html#E119:71_1_1"><i><font color="Green">E125</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>41</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:71_1_1_21"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E11:71_1_1_21"/><i><font color="Green">E134</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> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E47:71_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_21"><i><font color="Green">E133</font></i></a></i>;<br/><a NAME="E12:71_1_1_21"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>41</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>32</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E118:71_1_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="graph_2.html#E119:71_1_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_1_21"><i><font color="Green">E132</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_21"><i><font color="Green">E133</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:71_1_1_21"/><i><font color="Green">E136</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>10</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E116:71_1_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_21"><i><font color="Green">E133</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:71_1_1_21"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E10:71_1_1_21"><i><font color="Green">E133</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>42</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:71_1_1_21"/><i><font color="Green">E137</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>42</sub></font> &amp; <font color="Maroon">c<sub>42</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>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_2.html#E10:71_1_1_21"><i><font color="Green">E133</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</a></i>;<br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> = 
<font color="Maroon">c<sub>2</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_2.html#E15:71_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_2.html#E116:71_1_1"><i><font color="Green">E123</font></i></a>, <a class="txt" href="graph_2.html#E16:71_1_1_21"><i><font color="Green">E137</font></i></a>, <a class="ref" href="graph_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>40</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E51:71_1_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_2.html#E5:71_1_1_21"><i><font color="Green">E130</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_21"><i><font color="Green">E131</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_21"><i><font color="Green">E133</font></i></a>, <a class="txt" href="graph_2.html#E11:71_1_1_21"><i><font color="Green">E134</font></i></a>, <a class="txt" href="graph_2.html#E12:71_1_1_21"><i><font color="Green">E135</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:71_1_1_21"/>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_1_21"><i><font color="Green">E129</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_21"><i><font color="Green">E133</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_1_21"><i><font color="Green">E136</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E125:71_1_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</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>32</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E126:71_1_1"/><i><font color="Green">E129</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>35</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>34</sub></font>
 ;<br/><a NAME="E127:71_1_1"/><i><font color="Green">E130</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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</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="71_1_1_22"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>38</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_22_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>38</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>39</sub></font>, <font color="Maroon">c<sub>40</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_22_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>38</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>39</sub></font>,<font color="Maroon">c<sub>40</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="E4:71_1_1_22_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">c<sub>39</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>40</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E1:71_1_1_22_1"><i><font color="Green">E131</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_1_22_1"><i><font color="Green">E132</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>41</sub></font> = <font color="Maroon">c<sub>39</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:71_1_1_22_1"/><i><font color="Green">E134</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>41</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_1_22_1"><i><font color="Green">E133</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_1_22_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>39</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>33</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E111:71_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graph_2.html#E114:71_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="graph_2.html#E115:71_1_1"><i><font color="Green">E122</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_22_1"><i><font color="Green">E133</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:71_1_1_22_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>41</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>33</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E110:71_1_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graph_2.html#E114:71_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="graph_2.html#E126:71_1_1"><i><font color="Green">E129</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_22_1"><i><font color="Green">E134</font></i></a>, <a class="ref" href="graph_2.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E9:71_1_1_22_1"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>41</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>33</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E7:71_1_1_22_1"><i><font color="Green">E135</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:71_1_1_22_1"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">c<sub>39</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>21</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>33</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E123:71_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_1_22_1"><i><font color="Green">E135</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_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>41</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_1_22_1"><i><font color="Green">E133</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>42</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:71_1_1_22_1"/><i><font color="Green">E139</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>42</sub></font> &amp; <font color="Maroon">c<sub>42</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_2.html#E6:71_1_1_22_1"><i><font color="Green">E134</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>21</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>33</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font> = 
<font color="Maroon">c<sub>21</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>41</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E8:71_1_1_22_1"><i><font color="Green">E136</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_22_1"><i><font color="Green">E138</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>41</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E51:71_1_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_2.html#E123:71_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_1_22_1"><i><font color="Green">E136</font></i></a>, <a class="txt" href="graph_2.html#E9:71_1_1_22_1"><i><font color="Green">E137</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>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E13:71_1_1_22_1"><i><font color="Green">E139</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>40</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E127:71_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_22_1"><i><font color="Green">E133</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_1_22_1"><i><font color="Green">E139</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:71_1_1_22_1"/>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</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>33</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E3:71_1_1_22_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_22_1"><i><font color="Green">E138</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="E2:71_1_1_22"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</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>33</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>39</sub></font>, <font color="Maroon">c<sub>40</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_1_22"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>38</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>39</sub></font>,<font color="Maroon">c<sub>40</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="E5:71_1_1_22"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">c<sub>39</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>21</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>33</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</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>33</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E2:71_1_1_22"><i><font color="Green">E131</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_1_22"><i><font color="Green">E132</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_1_22"/><b>then </b><i><font color="Green">E134</font></i>: 
( <font color="Maroon">c<sub>39</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>33</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</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>41</sub></font> = <font color="Maroon">c<sub>39</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E8:71_1_1_22"/><i><font color="Green">E135</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E111:71_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graph_2.html#E114:71_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="graph_2.html#E115:71_1_1"><i><font color="Green">E122</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_22"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:71_1_1_22"/><i><font color="Green">E136</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>41</sub></font> &amp; <font color="Maroon">c<sub>41</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_2.html#E111:71_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="graph_2.html#E114:71_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="graph_2.html#E115:71_1_1"><i><font color="Green">E122</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_22"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:71_1_1_22"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>41</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>33</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E110:71_1_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="graph_2.html#E114:71_1_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="graph_2.html#E126:71_1_1"><i><font color="Green">E129</font></i></a>, <a class="ref" href="graph_2.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E11:71_1_1_22"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E6:71_1_1_22"><i><font color="Green">E134</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>42</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:71_1_1_22"/><i><font color="Green">E138</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>42</sub></font> &amp; <font color="Maroon">c<sub>42</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_2.html#E9:71_1_1_22"><i><font color="Green">E136</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</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" target="_self">-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> = 
<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>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E127:71_1_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_1_22"><i><font color="Green">E138</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</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>41</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E13:71_1_1_22"><i><font color="Green">E138</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>21</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>33</sub></font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E51:71_1_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_22"><i><font color="Green">E134</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_1_22"><i><font color="Green">E137</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">c<sub>40</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E5:71_1_1_22"><i><font color="Green">E133</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_1_22"><i><font color="Green">E134</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:71_1_1_22"/>
<font color="Maroon">c<sub>38</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_1_22"><i><font color="Green">E132</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_1_22"><i><font color="Green">E135</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E129:71_1_1"/><b>then </b><i><font color="Green">E131</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</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>33</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E130:71_1_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</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>32</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>33</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E47:71_1_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_2.html#E98:71_1_1"><i><font color="Green">E115</font></i></a>, <a class="txt" href="graph_2.html#E103:71_1_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><b>hence </b><a NAME="E131:71_1_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E55:71_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="graph_2.html#E121:71_1_1"><i><font color="Green">E126</font></i></a>, <a class="txt" href="graph_2.html#E123:71_1_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="graph_2.html#E125:71_1_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="graph_2.html#E129:71_1_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="graph_2.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2"><b>suppose </b></a><a NAME="E1:71_1_2"/><i><font color="Green">E68</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="E2:71_1_2"/><b>then </b><i><font color="Green">E69</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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:71_1_2"/><b>then </b><i><font color="Green">E70</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_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:71_1_2"/><i><font color="Green">E71</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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:71_1_2"/><i><font color="Green">E72</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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E2:71_1_2"><i><font color="Green">E69</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" target="_self">-cut</a> <font color="Maroon">c<sub>3</sub></font> as    <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_2.html#E3:71_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="graph_2.html#T44" target="_self">Th44</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E7:71_1_2"/><i><font color="Green">E73</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E2:71"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="graph_2.html#T45" target="_self">Th45</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="E9:71_1_2"/><i><font color="Green">E74</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="E10:71_1_2"/><i><font color="Green">E75</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="E11:71_1_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="71_1_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="E1:71_1_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="E3:71_1_2_1"/><i><font color="Green">E76</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="E4:71_1_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_2.html#E4:71_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_2_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_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_2.html#E3:71_1_2_1"><i><font color="Green">E76</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_2.html#E9:71_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E13:71_1_2"/><i><font color="Green">E76</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="71_1_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="E1:71_1_2_2"/><i><font color="Green">E77</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="E3:71_1_2_2"/><i><font color="Green">E78</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="E4:71_1_2_2"/><i><font color="Green">E79</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_2.html#E1:71_1_2_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_2_2"><i><font color="Green">E78</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="E6:71_1_2_2"/><i><font color="Green">E80</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_2.html#E9:71_1_2"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E7:71_1_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_2.html#E4:71_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:71_1_2_2"/><b>then </b><i><font color="Green">E81</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_2.html#E6:71_1_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:71_1_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_2.html#E9:71_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_2_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_2"><i><font color="Green">E80</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:71_1_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_2.html#E3:71_1_2_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_2_2"><i><font color="Green">E81</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" target="_self">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5" target="_self">Def5</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="E16:71_1_2"/><i><font color="Green">E77</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="E17:71_1_2"/><i><font color="Green">E78</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="E18:71_1_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="71_1_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="E1:71_1_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="E3:71_1_2_3"/><i><font color="Green">E79</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="E4:71_1_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_2.html#E4:71_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_2_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_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_2.html#E3:71_1_2_3"><i><font color="Green">E79</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_2.html#E16:71_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E20:71_1_2"/><i><font color="Green">E79</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="71_1_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="E1:71_1_2_4"/><i><font color="Green">E80</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="E3:71_1_2_4"/><i><font color="Green">E81</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="E4:71_1_2_4"/><i><font color="Green">E82</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_2.html#E1:71_1_2_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_2_4"><i><font color="Green">E81</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="E6:71_1_2_4"/><i><font color="Green">E83</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_2.html#E16:71_1_2"><i><font color="Green">E77</font></i></a></i>;<br/>
<a NAME="E7:71_1_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_2.html#E4:71_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:71_1_2_4"/><b>then </b><i><font color="Green">E84</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_2.html#E6:71_1_2_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:71_1_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_2.html#E16:71_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_2.html#E17:71_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_2_4"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_4"><i><font color="Green">E83</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:71_1_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_2.html#E3:71_1_2_4"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_4"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_2_4"><i><font color="Green">E84</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" target="_self">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5" target="_self">Def5</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E22:71_1_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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E1:71_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/><b>then </b><i><font color="Green">E80</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="xcmplx_0.html#K6">-</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="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1

;<br/><a NAME="E24:71_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</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="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E3:71_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E25:71_1_2"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E26:71_1_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="xcmplx_0.html#K6">-</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="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E28:71_1_2"/><i><font color="Green">E82</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="E29:71_1_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="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E23:71_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_2.html#E24:71_1_2"><i><font color="Green">E81</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="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E31:71_1_2"/><i><font color="Green">E83</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="E32:71_1_2"/><i><font color="Green">E84</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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E1:71_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E33:71_1_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" target="_self">-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_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/><b>then </b><i><font color="Green">E85</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" target="_self">-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="xcmplx_0.html#K6">-</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="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1

;<br/><a NAME="E35:71_1_2"/><b>then </b><i><font color="Green">E86</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E23:71_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E36:71_1_2"/><i><font color="Green">E87</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="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E23:71_1_2"><i><font color="Green">E80</font></i></a></i>;<br/><a NAME="E37:71_1_2"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E38:71_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E39:71_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</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="E40:71_1_2"/><i><font color="Green">E88</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_2.html#E36:71_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/><b>thus </b><a NAME="E41:71_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E2:71"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_2.html#E5:71_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="graph_2.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E42:71_1_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_2.html#E7:71_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="graph_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E43:71_1_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_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E40:71_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>thus </b><a NAME="E44:71_1_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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E1:71_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="graph_2.html#T12" target="_self">Th12</a></i>;<br/><b>thus </b><a NAME="E45:71_1_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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E1:71_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="graph_2.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E46:71_1_2"/><i><font color="Green">E89</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="71_1_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="E1:71_1_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="E3:71_1_2_7"/><i><font color="Green">E90</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="E4:71_1_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_2.html#E4:71_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_2_7"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_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_2.html#E3:71_1_2_7"><i><font color="Green">E90</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="E48:71_1_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="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E23:71_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_2.html#E24:71_1_2"><i><font color="Green">E81</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="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E50:71_1_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_2.html#E23:71_1_2"><i><font color="Green">E80</font></i></a></i>;<br/><b>then </b><i><font color="Green">E90</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="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>

;<br/><a NAME="E52:71_1_2"/><i><font color="Green">E91</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_2.html#E46:71_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E53:71_1_2"/><i><font color="Green">E92</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_2.html#E9:71_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E46:71_1_2"><i><font color="Green">E89</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="71_1_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="E1:71_1_2_9_1"/><i><font color="Green">E93</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="E3:71_1_2_9_1"/><i><font color="Green">E94</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="E4:71_1_2_9_1"/><i><font color="Green">E95</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_2.html#E1:71_1_2_9_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_2_9_1"><i><font color="Green">E94</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="E6:71_1_2_9_1"/><i><font color="Green">E96</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_2.html#E4:71_1_2_9_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_2_9_1"/><i><font color="Green">E97</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_2.html#E51:71_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_2.html#E52:71_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_2_9_1"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:71_1_2_9_1"/><i><font color="Green">E98</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="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E23:71_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E31:71_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_9_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="graph_2.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E9:71_1_2_9_1"/><i><font color="Green">E99</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="recdef_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_2.html#E7:71_1_2_9_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:71_1_2_9_1"/><b>then </b><i><font color="Green">E100</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_2.html#E53:71_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_2_9_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_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_2.html#E4:71_1_2_9_1"><i><font color="Green">E95</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="E13:71_1_2_9_1"/><i><font color="Green">E101</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_2.html#E6:71_1_2_9_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</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_2.html#E8:71_1_2_9_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_2_9_1"><i><font color="Green">E100</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_2.html#E13:71_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_2.html#E53:71_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_2_9_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="graph_2.html#E9:71_1_2_9_1"><i><font color="Green">E99</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_2.html#E13:71_1_2_9_1"><i><font color="Green">E101</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E1:71_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_2_9_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_2_9_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:71_1_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_2.html#E3:71_1_2_9_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_2_9_1"><i><font color="Green">E100</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="E2:71_1_2_9"/><i><font color="Green">E93</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="E4:71_1_2_9"/><i><font color="Green">E94</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="E5:71_1_2_9"/><i><font color="Green">E95</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_2.html#E2:71_1_2_9"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_2_9"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_2_9"/><b>then </b><i><font color="Green">E96</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="E8:71_1_2_9"/><i><font color="Green">E97</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_2.html#E51:71_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_2.html#E52:71_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_9"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:71_1_2_9"/><i><font color="Green">E98</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_2.html#E51:71_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_2.html#E52:71_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_9"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:71_1_2_9"/><b>then </b><i><font color="Green">E99</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="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E23:71_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E31:71_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="graph_2.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E11:71_1_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_2.html#E6:71_1_2_9"><i><font color="Green">E96</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="E13:71_1_2_9"/><i><font color="Green">E100</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_2.html#E9:71_1_2_9"><i><font color="Green">E98</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E1:71_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_2_9"><i><font color="Green">E100</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</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_2.html#E13:71_1_2_9"><i><font color="Green">E100</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="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E13:71_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_9"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_2_9"><i><font color="Green">E99</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_2.html#E5:71_1_2_9"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_9"><i><font color="Green">E96</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:71_1_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_2.html#E4:71_1_2_9"><i><font color="Green">E94</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_2_9"><i><font color="Green">E97</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E55:71_1_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="E56:71_1_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_2.html#E9:71_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E57:71_1_2"/><i><font color="Green">E93</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="71_1_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="E1:71_1_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="E3:71_1_2_10"/><i><font color="Green">E94</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="E4:71_1_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_2.html#E4:71_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_2_10"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_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_2.html#E3:71_1_2_10"><i><font color="Green">E94</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="E59:71_1_2"/>
1 <a href="xcmplx_0.html#K6">-</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E35:71_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E60:71_1_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" target="_self">-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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E62:71_1_2"/><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>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_2.html#E34:71_1_2"><i><font color="Green">E85</font></i></a></i>;<br/><a NAME="E63:71_1_2"/><i><font color="Green">E95</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/><i><font color="Green">E96</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E62:71_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="graph_2.html#T4" target="_self">Th4</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

;<br/><a NAME="E65:71_1_2"/><i><font color="Green">E97</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_2.html#E57:71_1_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E66:71_1_2"/><i><font color="Green">E98</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_2.html#E16:71_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_2.html#E57:71_1_2"><i><font color="Green">E93</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="71_1_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="E1:71_1_2_12_1"/><i><font color="Green">E99</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" target="_self">-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="E3:71_1_2_12_1"/><i><font color="Green">E100</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="E4:71_1_2_12_1"/><i><font color="Green">E101</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" target="_self">-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" target="_self">-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_2.html#E1:71_1_2_12_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_2_12_1"><i><font color="Green">E100</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="E6:71_1_2_12_1"/><i><font color="Green">E102</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_2_12_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_2_12_1"/><i><font color="Green">E103</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_2.html#E64:71_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_2.html#E65:71_1_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_2_12_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:71_1_2_12_1"/><i><font color="Green">E104</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="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E34:71_1_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graph_2.html#E63:71_1_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_12_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="graph_2.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E9:71_1_2_12_1"/><i><font color="Green">E105</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="recdef_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_2.html#E7:71_1_2_12_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:71_1_2_12_1"/><b>then </b><i><font color="Green">E106</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_2.html#E66:71_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_2_12_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_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_2.html#E4:71_1_2_12_1"><i><font color="Green">E101</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="E13:71_1_2_12_1"/><i><font color="Green">E107</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" target="_self">-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_2.html#E6:71_1_2_12_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</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_2.html#E8:71_1_2_12_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_2_12_1"><i><font color="Green">E106</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_2.html#E20:71_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_2.html#E66:71_1_2"><i><font color="Green">E98</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_2_12_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="graph_2.html#E9:71_1_2_12_1"><i><font color="Green">E105</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_2.html#E13:71_1_2_12_1"><i><font color="Green">E107</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E32:71_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_2_12_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_2_12_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:71_1_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_2.html#E3:71_1_2_12_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_2_12_1"><i><font color="Green">E106</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="E2:71_1_2_12"/><i><font color="Green">E99</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="E4:71_1_2_12"/><i><font color="Green">E100</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="E5:71_1_2_12"/><i><font color="Green">E101</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_2.html#E2:71_1_2_12"><i><font color="Green">E99</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_2_12"><i><font color="Green">E100</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_2_12"/><b>then </b><i><font color="Green">E102</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="E8:71_1_2_12"/><i><font color="Green">E103</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E64:71_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_2.html#E65:71_1_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_12"><i><font color="Green">E102</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:71_1_2_12"/><i><font color="Green">E104</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_2.html#E64:71_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="graph_2.html#E65:71_1_2"><i><font color="Green">E97</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_12"><i><font color="Green">E102</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:71_1_2_12"/><b>then </b><i><font color="Green">E105</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="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E34:71_1_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graph_2.html#E63:71_1_2"><i><font color="Green">E95</font></i></a>, <a class="ref" href="graph_2.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E11:71_1_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_2.html#E6:71_1_2_12"><i><font color="Green">E102</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="E13:71_1_2_12"/><i><font color="Green">E106</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" target="_self">-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_2.html#E9:71_1_2_12"><i><font color="Green">E104</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</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" target="_self">-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_2.html#E32:71_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_2_12"><i><font color="Green">E106</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</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_2.html#E13:71_1_2_12"><i><font color="Green">E106</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="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E20:71_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_12"><i><font color="Green">E102</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_2_12"><i><font color="Green">E105</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_2.html#E5:71_1_2_12"><i><font color="Green">E101</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_2_12"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:71_1_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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_2_12"><i><font color="Green">E100</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_2_12"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E68:71_1_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" target="_self">-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="E69:71_1_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_2.html#E16:71_1_2"><i><font color="Green">E77</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="71_1_3"><b>suppose </b></a><a NAME="E1:71_1_3"/><i><font color="Green">E68</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="E2:71_1_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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:71_1_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="E4:71_1_3"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:71_1_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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:71_1_3"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</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="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E5:71"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:71_1_3"/><i><font color="Green">E71</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" target="_self">-cut</a> <font color="Maroon">c<sub>3</sub></font> as    <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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="graph_2.html#T44" target="_self">Th44</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E9:71_1_3"/><i><font color="Green">E72</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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:71_1_3"/><i><font color="Green">E73</font></i>: 
1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E2:71"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_2.html#E10:71"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="graph_2.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E11:71_1_3"/><b>then </b><i><font color="Green">E74</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" target="_self">-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_2.html#D7" target="_self">Def7</a></i>;<br/><b>reconsider </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>  as    <a href="hidden.html#M1">set</a>  ;<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="E14:71_1_3"/><i><font color="Green">E75</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E15:71_1_3"/><i><font color="Green">E76</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> <font color="Maroon">c<sub>9</sub></font> 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="E16:71_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <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="71_1_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="E1:71_1_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<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="E3:71_1_3_1"/><i><font color="Green">E77</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="E4:71_1_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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_3_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_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_2.html#E3:71_1_3_1"><i><font color="Green">E77</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_2.html#E14:71_1_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E18:71_1_3"/><i><font color="Green">E77</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="71_1_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="E1:71_1_3_2"/><i><font color="Green">E78</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="E3:71_1_3_2"/><i><font color="Green">E79</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="E4:71_1_3_2"/><i><font color="Green">E80</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_2.html#E1:71_1_3_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_3_2"><i><font color="Green">E79</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="E6:71_1_3_2"/><i><font color="Green">E81</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_2.html#E14:71_1_3"><i><font color="Green">E75</font></i></a></i>;<br/>
<a NAME="E7:71_1_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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:71_1_3_2"/><b>then </b><i><font color="Green">E82</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_2.html#E6:71_1_3_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:71_1_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_2.html#E14:71_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_2.html#E15:71_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_3_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_2"><i><font color="Green">E81</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:71_1_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_2.html#E3:71_1_3_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_3_2"><i><font color="Green">E82</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" target="_self">FinSubsequence</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5" target="_self">Def5</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="E21:71_1_3"/><i><font color="Green">E78</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="E22:71_1_3"/><i><font color="Green">E79</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="E23:71_1_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="71_1_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="E1:71_1_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="E3:71_1_3_3"/><i><font color="Green">E80</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="E4:71_1_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_2.html#E9:71_1_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_3_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_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_2.html#E3:71_1_3_3"><i><font color="Green">E80</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_2.html#E21:71_1_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E25:71_1_3"/><i><font color="Green">E80</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="71_1_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="E1:71_1_3_4"/><i><font color="Green">E81</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="E3:71_1_3_4"/><i><font color="Green">E82</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="E4:71_1_3_4"/><i><font color="Green">E83</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_2.html#E1:71_1_3_4"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_3_4"><i><font color="Green">E82</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="E6:71_1_3_4"/><i><font color="Green">E84</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_2.html#E21:71_1_3"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E7:71_1_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_2.html#E9:71_1_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_4"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:71_1_3_4"/><b>then </b><i><font color="Green">E85</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_2.html#E6:71_1_3_4"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:71_1_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_2.html#E21:71_1_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_2.html#E22:71_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_3_4"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_4"><i><font color="Green">E84</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:71_1_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_2.html#E3:71_1_3_4"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_4"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_3_4"><i><font color="Green">E85</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" target="_self">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#D5" target="_self">Def5</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E27:71_1_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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>;<br/><b>then </b><i><font color="Green">E81</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="xcmplx_0.html#K6">-</a> 1

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="txt" href="graph_2.html#E8:71"><i><font color="Green">E65</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><b>hence </b><a NAME="E29:71_1_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_2.html#E6:71_1_3"><i><font color="Green">E70</font></i></a></i>;<br/><b>thus </b><a NAME="E30:71_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E2:71"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_2.html#E10:71"><i><font color="Green">E67</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="graph_2.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E31:71_1_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_2.html#E10:71_1_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="graph_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E32:71_1_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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E33:71_1_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_2.html#E9:71_1_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="graph_2.html#T12" target="_self">Th12</a></i>;<br/><b>thus </b><a NAME="E34:71_1_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_2.html#E4:71"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E1:71_1_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="graph_2.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E35:71_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <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="71_1_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="E1:71_1_3_6"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<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="E3:71_1_3_6"/><i><font color="Green">E82</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="E4:71_1_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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_3_6"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_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_2.html#E3:71_1_3_6"><i><font color="Green">E82</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> = <font color="Maroon">c<sub>9</sub></font> 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="E37:71_1_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="E38:71_1_3"/><b>then </b><i><font color="Green">E82</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="E39:71_1_3"/><b>then </b><i><font color="Green">E83</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="71_1_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="E1:71_1_3_7_1"/><i><font color="Green">E84</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="E3:71_1_3_7_1"/><i><font color="Green">E85</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="E4:71_1_3_7_1"/><i><font color="Green">E86</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_2.html#E1:71_1_3_7_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_3_7_1"><i><font color="Green">E85</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="E6:71_1_3_7_1"/><i><font color="Green">E87</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_2.html#E4:71_1_3_7_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_3_7_1"/><b>then </b><i><font color="Green">E88</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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E39:71_1_3"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:71_1_3_7_1"/><b>then </b><i><font color="Green">E89</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="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E38:71_1_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E39:71_1_3"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:71_1_3_7_1"/><i><font color="Green">E90</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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_7_1"><i><font color="Green">E87</font></i></a></i>;<br/><a NAME="E10:71_1_3_7_1"/><b>then </b><i><font color="Green">E91</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_2.html#E14:71_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_3_7_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_3_7_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_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_2.html#E4:71_1_3_7_1"><i><font color="Green">E86</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="E13:71_1_3_7_1"/><i><font color="Green">E92</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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_7_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</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_2.html#E8:71_1_3_7_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_7_1"><i><font color="Green">E91</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_2.html#E14:71_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_2.html#E18:71_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_2.html#E9:71_1_3_7_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_3_7_1"><i><font color="Green">E92</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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_3_7_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_3_7_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>
;<br/><b>hence </b><a NAME="E15:71_1_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_2.html#E3:71_1_3_7_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_7_1"><i><font color="Green">E91</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="E2:71_1_3_7"/><i><font color="Green">E84</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="E4:71_1_3_7"/><i><font color="Green">E85</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="E5:71_1_3_7"/><i><font color="Green">E86</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_2.html#E2:71_1_3_7"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_3_7"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_3_7"/><b>then </b><i><font color="Green">E87</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="E7:71_1_3_7"/><i><font color="Green">E88</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_2.html#E5:71_1_3_7"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:71_1_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_2.html#E39:71_1_3"><i><font color="Green">E83</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="E10:71_1_3_7"/><i><font color="Green">E89</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="E11:71_1_3_7"/><i><font color="Green">E90</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_2.html#E14:71_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_7"><i><font color="Green">E89</font></i></a></i>;<br/><a NAME="E12:71_1_3_7"/><i><font color="Green">E91</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="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E38:71_1_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="graph_2.html#E39:71_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_3_7"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_7"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:71_1_3_7"/><i><font color="Green">E92</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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_7"><i><font color="Green">E89</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:71_1_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_2.html#E10:71_1_3_7"><i><font color="Green">E89</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="E16:71_1_3_7"/><i><font color="Green">E93</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_2.html#E10:71_1_3_7"><i><font color="Green">E89</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</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_2.html#E9:71"><i><font color="Green">E66</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E16:71_1_3_7"><i><font color="Green">E93</font></i></a>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma8</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E18:71_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="graph_2.html#E5:71_1_3_7"><i><font color="Green">E86</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_7"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_7"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_2.html#E11:71_1_3_7"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_2.html#E12:71_1_3_7"><i><font color="Green">E91</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:71_1_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_2.html#E4:71_1_3_7"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_7"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_3_7"><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="E41:71_1_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="E42:71_1_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_2.html#E14:71_1_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E43:71_1_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="71_1_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="E1:71_1_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="E3:71_1_3_8"/><i><font color="Green">E84</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="E4:71_1_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_2.html#E9:71_1_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_3_8"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_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_2.html#E3:71_1_3_8"><i><font color="Green">E84</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="E45:71_1_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="E46:71_1_3"/><b>then </b><i><font color="Green">E84</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="E47:71_1_3"/><b>then </b><i><font color="Green">E85</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="71_1_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="E1:71_1_3_9_1"/><i><font color="Green">E86</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" target="_self">-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="E3:71_1_3_9_1"/><i><font color="Green">E87</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="E4:71_1_3_9_1"/><i><font color="Green">E88</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" target="_self">-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" target="_self">-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_2.html#E1:71_1_3_9_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="graph_2.html#E3:71_1_3_9_1"><i><font color="Green">E87</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="E6:71_1_3_9_1"/><i><font color="Green">E89</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_3_9_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_3_9_1"/><b>then </b><i><font color="Green">E90</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_2.html#E11:71_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E47:71_1_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:71_1_3_9_1"/><b>then </b><i><font color="Green">E91</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="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E46:71_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_2.html#E47:71_1_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:71_1_3_9_1"/><i><font color="Green">E92</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_2.html#E11:71_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_9_1"><i><font color="Green">E89</font></i></a></i>;<br/><a NAME="E10:71_1_3_9_1"/><b>then </b><i><font color="Green">E93</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_2.html#E21:71_1_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_3_9_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_2.html#E8:71_1_3_9_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_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_2.html#E4:71_1_3_9_1"><i><font color="Green">E88</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="E13:71_1_3_9_1"/><i><font color="Green">E94</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_2.html#E11:71_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_9_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</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_2.html#E8:71_1_3_9_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_9_1"><i><font color="Green">E93</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_2.html#E21:71_1_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_2.html#E25:71_1_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_2.html#E9:71_1_3_9_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_3_9_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>25</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E9:71_1_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_2.html#E11:71_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_3_9_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_3_9_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="graph_2.html#D1" target="_self">Def1</a></i>
;<br/><b>hence </b><a NAME="E15:71_1_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_2.html#E3:71_1_3_9_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_9_1"><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><b>assume </b><a NAME="E2:71_1_3_9"/><i><font color="Green">E86</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="E4:71_1_3_9"/><i><font color="Green">E87</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="E5:71_1_3_9"/><i><font color="Green">E88</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_2.html#E2:71_1_3_9"><i><font color="Green">E86</font></i></a>, <a class="txt" href="graph_2.html#E4:71_1_3_9"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_3_9"/><b>then </b><i><font color="Green">E89</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="E7:71_1_3_9"/><i><font color="Green">E90</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_2.html#E5:71_1_3_9"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:71_1_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_2.html#E47:71_1_3"><i><font color="Green">E85</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="E10:71_1_3_9"/><i><font color="Green">E91</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="E11:71_1_3_9"/><i><font color="Green">E92</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_2.html#E21:71_1_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_9"><i><font color="Green">E91</font></i></a></i>;<br/><a NAME="E12:71_1_3_9"/><i><font color="Green">E93</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="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E46:71_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="graph_2.html#E47:71_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="graph_2.html#E7:71_1_3_9"><i><font color="Green">E90</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_9"><i><font color="Green">E91</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:71_1_3_9"/><i><font color="Green">E94</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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E11:71_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_9"><i><font color="Green">E91</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:71_1_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_2.html#E10:71_1_3_9"><i><font color="Green">E91</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="E16:71_1_3_9"/><i><font color="Green">E95</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_2.html#E10:71_1_3_9"><i><font color="Green">E91</font></i></a>, <a class="ref" href="graph_2.html#T1" target="_self">Th1</a></i>;<br/><span class="p1">(<span class="default">1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2" target="_self">-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_2.html#E9:71_1_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="graph_2.html#E11:71_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="graph_2.html#E28:71_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="graph_2.html#E16:71_1_3_9"><i><font color="Green">E95</font></i></a>, <a class="ref" href="graph_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="graph_2.html#E25:71_1_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="graph_2.html#E5:71_1_3_9"><i><font color="Green">E88</font></i></a>, <a class="txt" href="graph_2.html#E6:71_1_3_9"><i><font color="Green">E89</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_9"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E11:71_1_3_9"><i><font color="Green">E92</font></i></a>, <a class="txt" href="graph_2.html#E12:71_1_3_9"><i><font color="Green">E93</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:71_1_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" target="_self">-cut</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="graph_2.html#E4:71_1_3_9"><i><font color="Green">E87</font></i></a>, <a class="txt" href="graph_2.html#E10:71_1_3_9"><i><font color="Green">E91</font></i></a>, <a class="txt" href="graph_2.html#E13:71_1_3_9"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E49:71_1_3"/><b>then </b>
1,<font color="Maroon">c<sub>4</sub></font> <a href="graph_2.html#K2" target="_self">-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="E50:71_1_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_2.html#E21:71_1_3"><i><font color="Green">E78</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>
