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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:80"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>1</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>1</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> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</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> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 ;<br/>

<a NAME="E2:80"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 
<b>by </b><i><a class="ref" href="jordan3.html#T73" target="_self">Th73</a></i>;<br/>
<a NAME="E3:80"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V2">unfolded</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V3">s.n.c.</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="topreal1.html#V1">special</a> )
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:80"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E4:80"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="topreal1.html#V2">unfolded</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="topreal1.html#V3">s.n.c.</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="topreal1.html#V1">special</a> )
 
<b>by </b><i><a class="txt" href="jordan3.html#E1:80"><i><font color="Green">E70</font></i></a>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E5:80"/><i><font color="Green">E74</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan3.html#E3:80"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:80"/><i><font color="Green">E75</font></i>: 
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="jordan3.html#E4:80"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:80"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></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>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E8:80"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <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>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan3.html#E4:80"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan3.html#E6:80"><i><font color="Green">E75</font></i></a>, <a class="ref" href="jordan3.html#T27" target="_self">Th27</a></i>;<br/>
<b>then </b><i><font color="Green">E78</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 2</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="jordan3.html#E4:80"><i><font color="Green">E73</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<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="real_1.html#K5">-</a> 1

;<br/>
<a NAME="E10:80"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <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="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan3.html#E4:80"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E11:80"/><i><font color="Green">E80</font></i>: 
<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"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</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="jordan3.html#E9:80"><i><font color="Green">E78</font></i></a></i>;<br/>
<i><font color="Green">E81</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="jordan3.html#E5:80"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan3.html#E5:80"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/>
<a NAME="E13:80"/>
<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="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan3.html#E4:80"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E14:80"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan3.html#E7:80"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan3.html#E9:80"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E15:80"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</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>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_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>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan3.html#T15" target="_self">Th15</a></i>
<br/>.= 
<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>
<b>by </b><i><a class="txt" href="jordan3.html#E4:80"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan3.html#E7:80"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan3.html#E8:80"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan3.html#E10:80"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan3.html#E11:80"><i><font color="Green">E80</font></i></a>, <a class="ref" href="jordan3.html#T31" target="_self">Th31</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</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="jordan3.html#E6:80"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/>
<b>hence </b><a NAME="E17:80"/>
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" target="_self">mid</a> <font color="Maroon">c<sub>2</sub></font>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="jordan3.html#R1" target="_self">is_S-Seq_joining</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</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="jordan3.html#E2:80"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan3.html#E12:80"><i><font color="Green">E81</font></i></a>, <a class="ref" href="jordan3.html#D3" target="_self">Def3</a></i>;<br/>


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