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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G</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</font> be  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font>, <font color="Maroon">c1</font> be  <a href="graph_1.html#V8">directed</a>  <a href="graph_1.html#M1">Chain</a> of <font color="Maroon">G</font>;<br/>







<b>assume </b><a NAME="E1:23"/><i><font color="Green">E34</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="graph_2.html#K1">-cut</a> <font color="Maroon">c</font> )
 ;<br/>

<b>set </b><font color="Maroon">mn1c</font> = <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">vsc</font> =  <a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font>;<br/>
<a NAME="E2:23"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:23"/><b>then </b><i><font color="Green">E42</font></i>: 
not <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E4:23"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="graph_2.html#D11">GRAPH_2:def 11</a></i>;<br/>
<a NAME="E5:23"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">c1</font>
 
<b>by </b><i>, <a class="ref" href="graph_2.html#T45">GRAPH_2:45</a></i>;<br/>
<a NAME="E6:23"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">c1</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:23"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</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="E8:23"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:23"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">c1</font> )
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E10:23"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 0</span>)</span>
 
<b>by </b><i>, <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/>
<a NAME="E11:23"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="graph_2.html#D7">GRAPH_2:def 7</a></i>;<br/>
<a NAME="E12:23"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">n</font> <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"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E13:23"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E14:23"/><b>then </b>
not  <a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E15:23"/><b>then </b>
not <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E16:23"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <span class="p2">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span></span>)</span> <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="E17:23"/><b>then </b>
( 0 <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">m</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <span class="p2">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span></span>)</span> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i/>;<br/>
<a NAME="E18:23"/><b>then </b>
<span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <span class="p2">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="graph_2.html#D1">GRAPH_2:def 1</a></i>;<br/>
<a NAME="E19:23"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <span class="p2">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E20:23"/>
 <a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2">-cut</a> <span class="p1">(<span class="default"><a href="graph_2.html#K7">vertex-seq</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="graph_2.html#D11">GRAPH_2:def 11</a></i>;<br/>


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