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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="topreal1.html#V1">special</a> <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>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="realset1.html#V1">trivial</a> <a href="topreal1.html#V1">special</a> <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="E1:29"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</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>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 ;<br/>

<a NAME="E2:29"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default">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> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_2.html#D2">GRAPH_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = 2,<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="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E3:29"/><i><font color="Green">E29</font></i>: 
1 <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="ref" href="realset1.html#T13">REALSET1:13</a></i>;<br/>
<a NAME="E4:29"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</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>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:29"/><b>then </b><i><font color="Green">E30</font></i>: 
2 <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="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E3:29"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</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">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> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default">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> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

<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="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="topreal8.html#E5:29"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topreal8.html#E13"><i><font color="Green">Lemma12</font></i></a></i>
;<br/>
<a NAME="E7:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default">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> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E8:29"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">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> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E3:29"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>then </b><i><font color="Green">E31</font></i>: <span class="p1">(<span class="default">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> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default">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> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</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> 1</span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E3:29"><i><font color="Green">E29</font></i></a>, <a class="ref" href="graph_2.html#T12">GRAPH_2:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E3:29"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/>
<a NAME="E10:29"/><i><font color="Green">E32</font></i>: 
2,<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="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font> is <a href="topreal1.html#V1">special</a>
 
<b>by </b><i><a class="ref" href="topreal8.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E11:29"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="txt" href="topreal8.html#E3:29"><i><font color="Green">E29</font></i></a>, <a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E12:29"/>
<font color="Maroon">c<sub>1</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>2</sub></font> is <a href="topreal1.html#V1">special</a>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:29"><i><font color="Green">E27</font></i></a>, <a class="txt" href="topreal8.html#E2:29"><i><font color="Green">E28</font></i></a>, <a class="txt" href="topreal8.html#E9:29"><i><font color="Green">E31</font></i></a>, <a class="txt" href="topreal8.html#E10:29"><i><font color="Green">E32</font></i></a>, <a class="ref" href="goboard2.html#T13">GOBOARD2:13</a></i>;<br/>


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