<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="msscyc_1.html#V2" target="_self">empty</a> <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green">E10</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<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>4</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="graph_2.html#T44">GRAPH_2:44</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="msscyc_1.html#E1:18"><i><font color="Green">E10</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E12</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</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>5</sub></font> )
 ;<br/><a NAME="E2:18_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green">E13</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</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>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:18_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="graph_2.html#T1">GRAPH_2:1</a></i>;<br/><a NAME="E5:18_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <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="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_1.html#E1:18_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E6:18_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_1.html#E4:18_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/><a NAME="E7:18_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</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="msscyc_1.html#E4:18_1"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E8:18_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</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>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:18"><i><font color="Green">E10</font></i></a>, <a class="txt" href="msscyc_1.html#E3:18"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msscyc_1.html#E1:18_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="msscyc_1.html#E4:18_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:18_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:18_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</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>4</sub></font> )
 <b>by </b><i><a class="txt" href="msscyc_1.html#E1:18"><i><font color="Green">E10</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/><b>hence </b><a NAME="E11:18_1"/>
the <a href="graph_1.html#U3">Source</a> of <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="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <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="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="msscyc_1.html#E5:18_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="msscyc_1.html#E6:18_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="msscyc_1.html#E7:18_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="graph_1.html#D12">GRAPH_1:def 12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">c<sub>4</sub></font> is  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="graph_1.html#D12">GRAPH_1:def 12</a></i>;<br/>


</div>
