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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="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>2</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/>

<a NAME="E2:20"/><i><font color="Green">E19</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:20"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="topreal8.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:20"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E5:20"/><b>then </b><i><font color="Green">E21</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="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E6:20"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topreal8.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="topreal8.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E7:20"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <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"><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="topreal8.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>;<br/>
<a NAME="E8:20"/><i><font color="Green">E23</font></i>: 
<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="binarith.html#K5">-'</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="jordan3.html#T7">JORDAN3:7</a></i>;<br/>
<a NAME="E9:20"/><i><font color="Green">E24</font></i>: 
<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> 1</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>
 
<b>by </b><i><a class="txt" href="topreal8.html#E5:20"><i><font color="Green">E21</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E10:20"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</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>3</sub></font></span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_2.html#D2">GRAPH_2:def 2</a></i>;<br/>
<a NAME="E11:20"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_5.html#K1">-:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K6">-|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="topreal8.html#E3:20"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_6.html#T44">FINSEQ_6:44</a></i>
<br/>.= 
<span class="p1">(<span class="default">1,<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>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="topreal8.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topreal8.html#E10:20"><i><font color="Green">E25</font></i></a>, <a class="ref" href="topreal8.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default">1,<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>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<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#E5:20"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topreal8.html#E7:20"><i><font color="Green">E22</font></i></a>, <a class="ref" href="graph_2.html#T6">GRAPH_2:6</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="topreal8.html#E8:20"><i><font color="Green">E23</font></i></a>, <a class="txt" href="topreal8.html#E9:20"><i><font color="Green">E24</font></i></a>, <a class="ref" href="graph_2.html#T9">GRAPH_2:9</a></i>
;<br/><br/>


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