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<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">vs</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:71"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c</font> is  not  <a href="graph_2.html#V3" target="_self">simple</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E2:71"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">vs</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c</font>
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:71"/><i><font color="Green">E34</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; ( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E5:71"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <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#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:71"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E8:71"/><i><font color="Green">E38</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E9:71"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E10:71"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1"/>
( ( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) or ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) or ( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1"><b>suppose </b></a><a NAME="E1:71_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 ;<br/><div class="add"><a NAME="E2:71_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:71_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:71_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:71_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:71_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:71_1_1"/><i><font color="Green">E49</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:71_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E31:71_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:71_1_1"/><b>then </b><i><font color="Green">E52</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="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:71_1_1"/><i><font color="Green">E53</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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</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="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">c1</font> = 1,<font color="Maroon">n1</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">c</font> as    <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, , <a class="txt" href="graph_2.html#E9:71_1_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c2</font> = <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">c</font> as    <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, <a class="txt" href="graph_2.html#E9:71_1_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>set </b><font color="Maroon">p2'</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>;<br/><b>reconsider </b><font color="Maroon">pp</font> = 1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> ;<br/><b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">m</font>,<span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font> ;<br/><a NAME="E15:71_1_1"/><i><font color="Green">E54</font></i>: 
( 1 <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">vs</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:71_1_1"/><i><font color="Green">E57</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> <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 &amp; <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 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E20:71_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="graph_2.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:71_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">pp</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c1</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E20:71_1_1"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, , <a class="txt" href="graph_2.html#E10:71_1_1"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E18:71_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">p2</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c2</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E20:71_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, <a class="txt" href="graph_2.html#E10:71_1_1"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E19:71_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pp</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E20:71_1_1"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E21:71_1_1"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E22:71_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E24:71_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E25:71_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <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#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E26:71_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> <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</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E27:71_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</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</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E28:71_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">lc21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E30:71_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">lc21</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span>
 ;<br/><a NAME="E31:71_1_1"/><i><font color="Green">E68</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</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">p2</font> )
 <b>by </b><i/>;<br/><a NAME="E32:71_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E85</font></i>: <font color="Maroon">p2</font> = 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font></span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">p2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>,1 <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">p2</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font></span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="graph_2.html#D1" target="_self">Def1</a>, </i>
;<br/><a NAME="E34:71_1_1"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E35:71_1_1"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">m</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, </i>;<br/><a NAME="E36:71_1_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>1,1 <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">p2</font> = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="graph_2.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 0</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, <a class="txt" href="graph_2.html#E33:71_1_1"><i><font color="Green">E85</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">m</font>,<font color="Maroon">m</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>
<b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, </i>
;<br/><a NAME="E38:71_1_1"/><b>then </b><i><font color="Green">E162</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> 2,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font></span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">p2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><b>set </b><font color="Maroon">domfc</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">fc</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E40:71_1_1"/><i><font color="Green">E163</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E41:71_1_1"/><i><font color="Green">E164</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) </span>}</span>  holds <br/><font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E42:71_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_3"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) )
 ;<br/>
<a NAME="E4:71_1_1_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</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="graph_2.html#T4" target="_self">Th4</a>, <a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fc</font> = <font color="Maroon">fc</font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E44:71_1_1"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">fc</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_1_4"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fc</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_4"/><i><font color="Green">E168</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:71_1_1_4"/><i><font color="Green">E169</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="graph_2.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:71_1_1_4"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</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="graph_2.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E7:71_1_1_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</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="graph_2.html#T4" target="_self">Th4</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:71_1_1_4"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:71_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 
<b>by </b><i><a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, , , </i>;<br/>
<b>hence </b><a NAME="E10:71_1_1_4"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T8" target="_self">Th8</a>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fc</font> = <font color="Maroon">fc</font> as    <a href="graph_2.html#M1" target="_self">FinSubsequence</a> of <font color="Maroon">c</font> <b>by </b><i><a class="txt" href="graph_2.html#E1:71"><i><font color="Green">E30</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">fc</font>
;<br/><b>set </b><font color="Maroon">domfvs</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">fvs</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E47:71_1_1"/><i><font color="Green">E195</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E48:71_1_1"/><i><font color="Green">E196</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) </span>}</span>  holds <br/><font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E49:71_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_5"/><i><font color="Green">E197</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) )
 ;<br/>
<a NAME="E4:71_1_1_5"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E5:71_1_1_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:71_1_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fvs</font> = <font color="Maroon">fvs</font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E51:71_1_1"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">fvs</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">vs</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_1_6"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fvs</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_6"/><i><font color="Green">E200</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:71_1_1_6"/><i><font color="Green">E201</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:71_1_1_6"/><i><font color="Green">E202</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) )
 <b>by </b><i/>;<br/>
<a NAME="E7:71_1_1_6"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E8:71_1_1_6"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:71_1_1_6"/><b>then </b><i><font color="Green">E203</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:71_1_1_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 
<b>by </b><i>, <a class="ref" href="graph_2.html#T8" target="_self">Th8</a>, , </i>;<br/>
<b>hence </b><a NAME="E11:71_1_1_6"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">vs</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fvs</font> = <font color="Maroon">fvs</font> as    <a href="graph_2.html#M1" target="_self">FinSubsequence</a> of <font color="Maroon">vs</font> <b>by </b><i><a class="txt" href="graph_2.html#E1:71"><i><font color="Green">E30</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">fvs</font>
;<br/><a NAME="E53:71_1_1"/><i><font color="Green">E204</font></i>: 
( <font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">pp</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">pp</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="graph_2.html#T5" target="_self">Th5</a>, </i>;<br/><b>then reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c2</font> as    <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, , , <a class="txt" href="graph_2.html#E15:71_1_1"><i><font color="Green">E54</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">c'</font>
;<br/><b>take </b><font color="Maroon">p1</font> = <font color="Maroon">pp</font> <a href="graph_2.html#K4" target="_self">^'</a> <font color="Maroon">p2</font>;<br/><a NAME="E55:71_1_1"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma90</font></i></a></i>;<br/><a NAME="E56:71_1_1"/><i><font color="Green">E219</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , , </i>;<br/><a NAME="E57:71_1_1"/><b>then </b><i><font color="Green">E220</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E58:71_1_1"/>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E59:71_1_1"/><b>then </b><i><font color="Green">E221</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="txt" href="graph_2.html#E27:71_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">c'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</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</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E61:71_1_1"/><i><font color="Green">E222</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/><b>thus </b><a NAME="E62:71_1_1"/>
<font color="Maroon">p1</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, , , , </i>;<br/><a NAME="E63:71_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <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#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E64:71_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E65:71_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E66:71_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pp</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E67:71_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E68:71_1_1"/>
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E69:71_1_1"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">p2</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">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <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> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E70:71_1_1"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</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</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E71:71_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E72:71_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E73:71_1_1"/>
<font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</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">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, , </i>;<br/><b>set </b><font color="Maroon">DL</font> = <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">DR</font> = <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_8_1"/><i><font color="Green">E224</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) )
 ;<br/><a NAME="E4:71_1_1_8_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:71_1_1_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:71_1_1_8"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_1_8_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_8_2_1"><b>suppose </b></a><a NAME="E1:71_1_1_8_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_8_2_1"/><i><font color="Green">E252</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> )
 ;<br/><b>thus </b><a NAME="E4:71_1_1_8_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_8_2_2"><b>suppose </b></a><a NAME="E1:71_1_1_8_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_8_2_2"/><i><font color="Green">E253</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> )
 ;<br/><b>thus </b><a NAME="E4:71_1_1_8_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E75:71_1_1"/><b>then </b><i><font color="Green">E254</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E76:71_1_1"/><i><font color="Green">E255</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> &amp; <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_9"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:71_1_1_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_9_1"/><i><font color="Green">E256</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> )
 ;<br/><a NAME="E4:71_1_1_9_1"/>
<font color="Maroon">k</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/><b>hence </b><a NAME="E5:71_1_1_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:71_1_1_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:71_1_1_9"/><i><font color="Green">E257</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> )
 ;<br/>
<a NAME="E5:71_1_1_9"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:71_1_1_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">DL</font> = <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon">DR</font> = <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</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">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:71_1_1_10"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DL</font>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_10"/><i><font color="Green">E258</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> )
 ;<br/><b>assume </b><a NAME="E4:71_1_1_10"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DR</font>
 ;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:71_1_1_10"/><i><font color="Green">E259</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E7:71_1_1_10"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:71_1_1_10"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E9:71_1_1_10"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E80:71_1_1"/><b>then </b><i><font color="Green">E260</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon">DL</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">DR</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/><a NAME="E81:71_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">lc21</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E261</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">DR</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font>

;<br/><a NAME="E83:71_1_1"/><i><font color="Green">E262</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">DL</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">DR</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E84:71_1_1"/>
<font color="Maroon">DL</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E85:71_1_1"/><b>then </b><i><font color="Green">E263</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font> <a href="hidden.html#R1">=</a>  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54">FINSEQ_3:54</a></i>;<br/><a NAME="E86:71_1_1"/><b>then </b><i><font color="Green">E264</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><a NAME="E87:71_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">DL</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E88:71_1_1"/><b>then </b><i><font color="Green">E265</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E89:71_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">DR</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E90:71_1_1"/><b>then </b><i><font color="Green">E266</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>set </b><font color="Maroon">SL</font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font>;<br/><b>set </b><font color="Maroon">SR</font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_12_1"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_12_1"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:71_1_1_12_1"/><i><font color="Green">E271</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="graph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:71_1_1_12_1"/><i><font color="Green">E272</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_1_12_1"/><b>then </b><i><font color="Green">E273</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="graph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:71_1_1_12_1"/><b>then </b><i><font color="Green">E274</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:71_1_1_12_1"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="graph_2.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E10:71_1_1_12_1"/><b>then </b><i><font color="Green">E276</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_1_12_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_1_12_1"/><i><font color="Green">E277</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , <a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="ref" href="graph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, <a class="txt" href="graph_2.html#E24"><i><font color="Green">Lemma119</font></i></a>, <a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, , , , <a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E15:71_1_1_12_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="graph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:71_1_1_12"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_1_12"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:71_1_1_12"/><i><font color="Green">E280</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="graph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_1_12"/><b>then </b><i><font color="Green">E281</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E7:71_1_1_12"/><i><font color="Green">E282</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:71_1_1_12"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="graph_2.html#T12" target="_self">Th12</a>, <a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:71_1_1_12"/><i><font color="Green">E283</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> )
 ;<br/><a NAME="E11:71_1_1_12"/><i><font color="Green">E284</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="graph_2.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E12:71_1_1_12"/><i><font color="Green">E285</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T12" target="_self">Th12</a>, , <a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:71_1_1_12"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font>
 <b>by </b><i>, , <a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:71_1_1_12"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="graph_2.html#D3" target="_self">Def3</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:71_1_1_12"/><i><font color="Green">E287</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, </i>;<br/><font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , , , <a class="txt" href="graph_2.html#E27"><i><font color="Green">Lemma122</font></i></a>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, <a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, <a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="txt" href="graph_2.html#E24"><i><font color="Green">Lemma119</font></i></a>, <a class="ref" href="graph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:71_1_1_12"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E92:71_1_1"/><b>then </b><i><font color="Green">E288</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_13"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_13_1"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_13_1"/><i><font color="Green">E290</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:71_1_1_13_1"/><i><font color="Green">E291</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:71_1_1_13_1"/><i><font color="Green">E292</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_1_13_1"/><i><font color="Green">E293</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T11" target="_self">Th11</a>, <a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:71_1_1_13_1"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="txt" href="graph_2.html#E32:71_1_1"><i><font color="Green">E84</font></i></a>, , </i>;<br/><a NAME="E9:71_1_1_13_1"/><i><font color="Green">E295</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:71_1_1_13_1"/><b>then </b><i><font color="Green">E296</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_1_13_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_1_13_1"/><i><font color="Green">E297</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E24"><i><font color="Green">Lemma119</font></i></a>, <a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<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> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, , <a class="txt" href="graph_2.html#E24"><i><font color="Green">Lemma119</font></i></a>, <a class="ref" href="graph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a>, <a class="txt" href="graph_2.html#E27"><i><font color="Green">Lemma122</font></i></a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, <a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, <a class="txt" href="graph_2.html#E27"><i><font color="Green">Lemma122</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E15:71_1_1_13_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:71_1_1_13"/><i><font color="Green">E298</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_1_13"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:71_1_1_13"/><i><font color="Green">E300</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T15" target="_self">Th15</a>, <a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_1_13"/><b>then </b><i><font color="Green">E301</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E8:71_1_1_13"/><i><font color="Green">E302</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c2</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T11" target="_self">Th11</a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:71_1_1_13"/><i><font color="Green">E303</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> )
 <b>by </b><i>, <a class="ref" href="graph_2.html#T11" target="_self">Th11</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:71_1_1_13"/><b>then </b><i><font color="Green">E304</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="txt" href="graph_2.html#E32:71_1_1"><i><font color="Green">E84</font></i></a>, </i>;<br/><a NAME="E11:71_1_1_13"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_1_13"/><i><font color="Green">E305</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_2.html#E24"><i><font color="Green">Lemma119</font></i></a>, </i>;<br/><font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, <a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, </i>
<br/>.= 
<font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, <a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, , <a class="ref" href="graph_2.html#T19" target="_self">Th19</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:71_1_1_13"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E94:71_1_1"/><b>then </b><i><font color="Green">E306</font></i>: 
<font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E95:71_1_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, , , <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><b>hence </b><a NAME="E96:71_1_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T13" target="_self">Th13</a>, , <a class="ref" href="graph_2.html#T14" target="_self">Th14</a>, <a class="ref" href="graph_2.html#T15" target="_self">Th15</a>, </i>;<br/><b>set </b><font color="Maroon">DL</font> = <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">DR</font> = <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_14_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_14_1"/><i><font color="Green">E307</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) )
 ;<br/><a NAME="E4:71_1_1_14_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="graph_2.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E5:71_1_1_14_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:71_1_1_14"/><i><font color="Green">E308</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_14_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_1_14_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span>  )
 </span><b>by </b><i><a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_14_2_1"><b>suppose </b></a><a NAME="E1:71_1_1_14_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_14_2_1"/><i><font color="Green">E309</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/><b>thus </b><a NAME="E4:71_1_1_14_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="graph_2.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_14_2_2"><b>suppose </b></a><a NAME="E1:71_1_1_14_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_14_2_2"/><i><font color="Green">E310</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 ;<br/><b>thus </b><a NAME="E4:71_1_1_14_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="graph_2.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E98:71_1_1"/><b>then </b><i><font color="Green">E311</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E99:71_1_1"/><i><font color="Green">E312</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> &amp; <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_15"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:71_1_1_15_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_15_1"/><i><font color="Green">E313</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E4:71_1_1_15_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:71_1_1_15_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:71_1_1_15"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:71_1_1_15"/><i><font color="Green">E314</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 ;<br/>
<a NAME="E5:71_1_1_15"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E6:71_1_1_15"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:71_1_1_15"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">DL</font> = <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon">DR</font> = <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_16"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</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">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:71_1_1_16"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DL</font>
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_1_16"/><i><font color="Green">E315</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/><b>assume </b><a NAME="E4:71_1_1_16"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DR</font>
 ;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:71_1_1_16"/><i><font color="Green">E316</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 ;<br/><a NAME="E7:71_1_1_16"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:71_1_1_16"/><b>then </b><i><font color="Green">E317</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:71_1_1_16"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E10:71_1_1_16"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="txt" href="graph_2.html#E24"><i><font color="Green">Lemma119</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E103:71_1_1"/><b>then </b><i><font color="Green">E318</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon">DL</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">DR</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/><a NAME="E104:71_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E105:71_1_1"/><b>then </b>
1 <a href="xcmplx_0.html#K6">-</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">p2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E106:71_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">lp21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E108:71_1_1"/>
<font color="Maroon">lp21</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">lp21</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">lp22</font> = <font color="Maroon">lp21</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><i><font color="Green">E321</font></i>: <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">lp22</font> = 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">lp21</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>
<br/>.= 
<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

;<br/><b>then </b><i><font color="Green">E322</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">DR</font> = 
<span class="p1">(<span class="default"><font color="Maroon">lp21</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, </i>
<br/>.= 
<font color="Maroon">lp21</font>

;<br/><a NAME="E112:71_1_1"/><i><font color="Green">E323</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</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/><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="graph_2.html#T19" target="_self">Th19</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"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>

;<br/><a NAME="E114:71_1_1"/><b>then </b><i><font color="Green">E324</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E115:71_1_1"/><i><font color="Green">E325</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">DL</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">DR</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E116:71_1_1"/><i><font color="Green">E326</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E117:71_1_1"/>
<font color="Maroon">DL</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E118:71_1_1"/><b>then </b><i><font color="Green">E327</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font> <a href="hidden.html#R1">=</a>  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54">FINSEQ_3:54</a></i>;<br/><a NAME="E119:71_1_1"/><b>then </b><i><font color="Green">E328</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><a NAME="E120:71_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">DL</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E121:71_1_1"/><b>then </b><i><font color="Green">E329</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E122:71_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">DR</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E123:71_1_1"/><b>then </b><i><font color="Green">E330</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>set </b><font color="Maroon">SL</font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font>;<br/><b>set </b><font color="Maroon">SR</font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_20"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_20_1"/><i><font color="Green">E331</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">pp</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_20_1"/><i><font color="Green">E332</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:71_1_1_20_1"/><i><font color="Green">E333</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pp</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T20" target="_self">Th20</a>, <a class="ref" href="graph_2.html#T21" target="_self">Th21</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:71_1_1_20_1"/><i><font color="Green">E334</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pp</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T22" target="_self">Th22</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_1_20_1"/><b>then </b><i><font color="Green">E335</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#D4" target="_self">Def4</a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:71_1_1_20_1"/><b>then </b><i><font color="Green">E336</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E29"><i><font color="Green">Lemma124</font></i></a>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:71_1_1_20_1"/><i><font color="Green">E337</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) ) </span>}</span> 
 <b>by </b><i><a class="ref" href="graph_2.html#D4" target="_self">Def4</a>, <a class="ref" href="graph_2.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E10:71_1_1_20_1"/><b>then </b><i><font color="Green">E338</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_1_20_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T22" target="_self">Th22</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_1_20_1"/><i><font color="Green">E339</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="graph_2.html#D4" target="_self">Def4</a>, <a class="ref" href="graph_2.html#T23" target="_self">Th23</a>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="graph_2.html#D5" target="_self">Def5</a>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="graph_2.html#D4" target="_self">Def4</a>, <a class="ref" href="graph_2.html#T22" target="_self">Th22</a>, , </i>
;<br/><b>hence </b><a NAME="E15:71_1_1_20_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T21" target="_self">Th21</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:71_1_1_20"/><i><font color="Green">E340</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_1_20"/><i><font color="Green">E341</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:71_1_1_20"/><i><font color="Green">E342</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T20" target="_self">Th20</a>, <a class="ref" href="graph_2.html#T21" target="_self">Th21</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_1_20"/><b>then </b><i><font color="Green">E343</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E7:71_1_1_20"/><i><font color="Green">E344</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T22" target="_self">Th22</a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:71_1_1_20"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:71_1_1_20"/><i><font color="Green">E345</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E11:71_1_1_20"/><i><font color="Green">E346</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:71_1_1_20"/><i><font color="Green">E347</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E29"><i><font color="Green">Lemma124</font></i></a>, , , , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:71_1_1_20"/><i><font color="Green">E348</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">pp</font>
 <b>by </b><i><a class="ref" href="graph_2.html#D4" target="_self">Def4</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:71_1_1_20"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:71_1_1_20"/><i><font color="Green">E349</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, </i>;<br/><font color="Maroon">pp</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="graph_2.html#D4" target="_self">Def4</a>, <a class="ref" href="graph_2.html#T24" target="_self">Th24</a>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="graph_2.html#T22" target="_self">Th22</a>, <a class="ref" href="graph_2.html#T23" target="_self">Th23</a>, , <a class="ref" href="graph_2.html#D5" target="_self">Def5</a>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:71_1_1_20"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">pp</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T21" target="_self">Th21</a>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E125:71_1_1"/><b>then </b><i><font color="Green">E350</font></i>: 
<font color="Maroon">pp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E126:71_1_1"/><i><font color="Green">E351</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">lp22</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">lp21</font>
 ;<br/><a NAME="E127:71_1_1"/><i><font color="Green">E352</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="graph_2.html#E24:71_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_21"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_21_1"/><i><font color="Green">E353</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_1_21_1"/><i><font color="Green">E354</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:71_1_1_21_1"/><i><font color="Green">E355</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T23" target="_self">Th23</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:71_1_1_21_1"/><i><font color="Green">E356</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_1_21_1"/><i><font color="Green">E357</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E24"><i><font color="Green">Lemma119</font></i></a>, <a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, <a class="txt" href="graph_2.html#E27"><i><font color="Green">Lemma122</font></i></a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:71_1_1_21_1"/><i><font color="Green">E358</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, <a class="ref" href="graph_2.html#T21" target="_self">Th21</a>, <a class="ref" href="graph_2.html#D5" target="_self">Def5</a>, </i>;<br/><a NAME="E9:71_1_1_21_1"/><i><font color="Green">E359</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:71_1_1_21_1"/><b>then </b><i><font color="Green">E360</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_1_21_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_1_21_1"/><i><font color="Green">E361</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</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"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="graph_2.html#D5" target="_self">Def5</a>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">fvs</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> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="graph_2.html#T24" target="_self">Th24</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a>, <a class="ref" href="graph_2.html#T26" target="_self">Th26</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="graph_2.html#T22" target="_self">Th22</a>, , <a class="ref" href="graph_2.html#T26" target="_self">Th26</a>, </i>
;<br/><b>hence </b><a NAME="E15:71_1_1_21_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:71_1_1_21"/><i><font color="Green">E362</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_1_21"/><i><font color="Green">E363</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:71_1_1_21"/><i><font color="Green">E364</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T23" target="_self">Th23</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_1_21"/><b>then </b><i><font color="Green">E365</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E8:71_1_1_21"/><i><font color="Green">E366</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E24"><i><font color="Green">Lemma119</font></i></a>, <a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, <a class="txt" href="graph_2.html#E27"><i><font color="Green">Lemma122</font></i></a>, <a class="ref" href="graph_2.html#D5" target="_self">Def5</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:71_1_1_21"/><i><font color="Green">E367</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_2.html#E24"><i><font color="Green">Lemma119</font></i></a>, <a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, <a class="txt" href="graph_2.html#E27"><i><font color="Green">Lemma122</font></i></a>, <a class="ref" href="graph_2.html#D5" target="_self">Def5</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:71_1_1_21"/><b>then </b><i><font color="Green">E368</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="graph_2.html#D3" target="_self">Def3</a>, <a class="txt" href="graph_2.html#E26"><i><font color="Green">Lemma121</font></i></a>, <a class="ref" href="graph_2.html#T21" target="_self">Th21</a>, </i>;<br/><a NAME="E11:71_1_1_21"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="graph_2.html#D5" target="_self">Def5</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_1_21"/><i><font color="Green">E369</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</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"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_2.html#E26:71_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="graph_2.html#T22" target="_self">Th22</a>, , </i>
<br/>.= 
<font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="graph_2.html#D5" target="_self">Def5</a>, <a class="ref" href="graph_2.html#T24" target="_self">Th24</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="graph_2.html#D5" target="_self">Def5</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:71_1_1_21"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E129:71_1_1"/><b>then </b><i><font color="Green">E370</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E130:71_1_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fvs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DL</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">DR</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T16" target="_self">Th16</a>, , <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><b>hence </b><a NAME="E131:71_1_1"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fvs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>by </b><i>, , , <a class="ref" href="graph_2.html#T20" target="_self">Th20</a>, <a class="ref" href="graph_2.html#T23" target="_self">Th23</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2"><b>suppose </b></a><a NAME="E1:71_1_2"/><i><font color="Green">E371</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 ;<br/><div class="add"><a NAME="E2:71_1_2"/><b>then </b><i><font color="Green">E372</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:71_1_2"/><b>then </b><i><font color="Green">E373</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="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:71_1_2"/><i><font color="Green">E374</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:71_1_2"/><i><font color="Green">E375</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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</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="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">c2</font> = <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">c</font> as    <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, , </i>;<br/><b>set </b><font color="Maroon">p2</font> = <font color="Maroon">m</font>,<span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>;<br/><a NAME="E7:71_1_2"/><i><font color="Green">E376</font></i>: 
<font color="Maroon">m</font>,<span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c2</font>
 <b>by </b><i>, , , </i>;<br/><b>set </b><font color="Maroon">domfc</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">fc</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:71_1_2"/><i><font color="Green">E377</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E10:71_1_2"/><i><font color="Green">E378</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span>  holds <br/><font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E11:71_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_2_1"/><i><font color="Green">E379</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> )
 ;<br/>
<a NAME="E4:71_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</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="graph_2.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fc</font> = <font color="Maroon">fc</font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E13:71_1_2"/><i><font color="Green">E380</font></i>: 
<font color="Maroon">fc</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_2_2"/><i><font color="Green">E381</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fc</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_2_2"/><i><font color="Green">E382</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:71_1_2_2"/><i><font color="Green">E383</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:71_1_2_2"/><i><font color="Green">E384</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</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="graph_2.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E7:71_1_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</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="E8:71_1_2_2"/><b>then </b><i><font color="Green">E385</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:71_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 
<b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, , , </i>;<br/>
<b>hence </b><a NAME="E10:71_1_2_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fc</font> = <font color="Maroon">fc</font> as    <a href="graph_2.html#M1" target="_self">FinSubsequence</a> of <font color="Maroon">c</font> <b>by </b><i><a class="txt" href="graph_2.html#E1:71"><i><font color="Green">E30</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">fc</font>
;<br/><b>set </b><font color="Maroon">domfvs</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">fvs</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E16:71_1_2"/><i><font color="Green">E386</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E17:71_1_2"/><i><font color="Green">E387</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span>  holds <br/><font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E18:71_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_2_3"/><i><font color="Green">E388</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 ;<br/>
<a NAME="E4:71_1_2_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fvs</font> = <font color="Maroon">fvs</font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E20:71_1_2"/><i><font color="Green">E389</font></i>: 
<font color="Maroon">fvs</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">vs</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_2_4"/><i><font color="Green">E390</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fvs</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_2_4"/><i><font color="Green">E391</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:71_1_2_4"/><i><font color="Green">E392</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:71_1_2_4"/><i><font color="Green">E393</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:71_1_2_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:71_1_2_4"/><b>then </b><i><font color="Green">E394</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:71_1_2_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E10:71_1_2_4"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">vs</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fvs</font> = <font color="Maroon">fvs</font> as    <a href="graph_2.html#M1" target="_self">FinSubsequence</a> of <font color="Maroon">vs</font> <b>by </b><i><a class="txt" href="graph_2.html#E1:71"><i><font color="Green">E30</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">fvs</font>
;<br/><b>take </b><font color="Maroon">c1</font> = <font color="Maroon">c2</font>;<br/><b>take </b><font color="Maroon">p1</font> = <font color="Maroon">m</font>,<span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>;<br/><a NAME="E22:71_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <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>, , , </i>;<br/><a NAME="E23:71_1_2"/><b>then </b><i><font color="Green">E395</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> <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</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E24:71_1_2"/><i><font color="Green">E396</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</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</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E25:71_1_2"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E26:71_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E28:71_1_2"/><i><font color="Green">E397</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E29:71_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">lc21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E31:71_1_2"/><i><font color="Green">E398</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">lc21</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span>
 ;<br/><a NAME="E32:71_1_2"/><i><font color="Green">E399</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> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <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 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E33:71_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<span class="p3">(<span class="default"><span class="p4">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E34:71_1_2"/><b>then </b><i><font color="Green">E400</font></i>: 
 <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"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E35:71_1_2"/><b>then </b><i><font color="Green">E401</font></i>: 
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">m</font>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E36:71_1_2"/><i><font color="Green">E402</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> <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="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E37:71_1_2"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E38:71_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E39:71_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><b>hence </b><a NAME="E40:71_1_2"/><i><font color="Green">E403</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="real_2.html#T173">REAL_2:173</a></i>;<br/><b>thus </b><a NAME="E41:71_1_2"/>
<font color="Maroon">p1</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c1</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E42:71_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E43:71_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>thus </b><a NAME="E44:71_1_2"/>
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, , , </i>;<br/><b>thus </b><a NAME="E45:71_1_2"/>
<font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</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">p1</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E46:71_1_2"/><i><font color="Green">E404</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_2_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_2_5"/><i><font color="Green">E405</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> )
 ;<br/>
<a NAME="E4:71_1_2_5"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_2_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">domfc</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E48:71_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">lc21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E50:71_1_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">lc21</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E406</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">domfc</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font>

;<br/><a NAME="E52:71_1_2"/><i><font color="Green">E407</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">domfc</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E53:71_1_2"/><i><font color="Green">E408</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>set </b><font color="Maroon">SR</font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_2_7_1"/><i><font color="Green">E409</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_2_7_1"/><i><font color="Green">E410</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:71_1_2_7_1"/><i><font color="Green">E411</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:71_1_2_7_1"/><i><font color="Green">E412</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_2_7_1"/><i><font color="Green">E413</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="graph_2.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:71_1_2_7_1"/><i><font color="Green">E414</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E9:71_1_2_7_1"/><i><font color="Green">E415</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:71_1_2_7_1"/><b>then </b><i><font color="Green">E416</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_2_7_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_2_7_1"/><i><font color="Green">E417</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<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> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="graph_2.html#T5" target="_self">Th5</a>, , , , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, , , <a class="ref" href="graph_2.html#T8" target="_self">Th8</a>, , </i>
;<br/><b>hence </b><a NAME="E15:71_1_2_7_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:71_1_2_7"/><i><font color="Green">E418</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_2_7"/><i><font color="Green">E419</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:71_1_2_7"/><i><font color="Green">E420</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_2_7"/><b>then </b><i><font color="Green">E421</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E8:71_1_2_7"/><i><font color="Green">E422</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c2</font>
 <b>by </b><i><a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:71_1_2_7"/><i><font color="Green">E423</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> )
 <b>by </b><i><a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:71_1_2_7"/><b>then </b><i><font color="Green">E424</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E11:71_1_2_7"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_2_7"/><i><font color="Green">E425</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, </i>;<br/><font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="graph_2.html#T5" target="_self">Th5</a>, , , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="graph_2.html#T8" target="_self">Th8</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:71_1_2_7"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E55:71_1_2"/><b>then </b>
<font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E56:71_1_2"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E57:71_1_2"/><i><font color="Green">E426</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_2_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_2_8"/><i><font color="Green">E427</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 ;<br/>
<a NAME="E4:71_1_2_8"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_2_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">domfvs</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E59:71_1_2"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<span class="p3">(<span class="default"><span class="p4">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E60:71_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<span class="p3">(<span class="default"><span class="p4">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<span class="p3">(<span class="default"><span class="p4">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">lp21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">m</font>,<span class="p3">(<span class="default"><span class="p4">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E62:71_1_2"/><i><font color="Green">E428</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">lp21</font> <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/>;<br/><a NAME="E63:71_1_2"/><i><font color="Green">E429</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">lp21</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <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"><font color="Maroon">m</font>,<span class="p3">(<span class="default"><span class="p4">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span>
 ;<br/><i><font color="Green">E430</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">domfvs</font> = 
<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">m</font>,<span class="p4">(<span class="default"><span class="p5">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i>, , </i>
<br/>.= 
 <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"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span>

;<br/><a NAME="E65:71_1_2"/><i><font color="Green">E431</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">domfvs</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/><a NAME="E66:71_1_2"/><i><font color="Green">E432</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>set </b><font color="Maroon">SR</font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_2_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_2_10_1"/><i><font color="Green">E433</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_2_10_1"/><i><font color="Green">E434</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:71_1_2_10_1"/><i><font color="Green">E435</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:71_1_2_10_1"/><i><font color="Green">E436</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <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">m</font>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_2_10_1"/><i><font color="Green">E437</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E8:71_1_2_10_1"/><i><font color="Green">E438</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="graph_2.html#T8" target="_self">Th8</a>, , </i>;<br/><a NAME="E9:71_1_2_10_1"/><i><font color="Green">E439</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:71_1_2_10_1"/><b>then </b><i><font color="Green">E440</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_2_10_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_2_10_1"/><i><font color="Green">E441</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</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">m</font>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">vs</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> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, , , </i>
;<br/><b>hence </b><a NAME="E15:71_1_2_10_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:71_1_2_10"/><i><font color="Green">E442</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_2_10"/><i><font color="Green">E443</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:71_1_2_10"/><i><font color="Green">E444</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_2_10"/><b>then </b><i><font color="Green">E445</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E8:71_1_2_10"/><i><font color="Green">E446</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">m</font>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E9:71_1_2_10"/><i><font color="Green">E447</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <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">m</font>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:71_1_2_10"/><b>then </b><i><font color="Green">E448</font></i>: 
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="graph_2.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E11:71_1_2_10"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_2_10"/><i><font color="Green">E449</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</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">m</font>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">m</font>,<span class="p2">(<span class="default"><span class="p3">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E15:71_1_2_10"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E68:71_1_2"/><b>then </b>
<font color="Maroon">m</font>,<span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E69:71_1_2"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fvs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_3"><b>suppose </b></a><a NAME="E1:71_1_3"/><i><font color="Green">E450</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 ;<br/><div class="add"><a NAME="E2:71_1_3"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:71_1_3"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:71_1_3"/><b>then </b><i><font color="Green">E451</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:71_1_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:71_1_3"/><b>then </b><i><font color="Green">E452</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</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</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:71_1_3"/><i><font color="Green">E453</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon">c1</font> = 1,<font color="Maroon">n1</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">c</font> as    <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, , </i>;<br/><b>set </b><font color="Maroon">pp</font> = 1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>;<br/><a NAME="E9:71_1_3"/><i><font color="Green">E454</font></i>: 
( 1 <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">vs</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:71_1_3"/><i><font color="Green">E455</font></i>: 
1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c1</font>
 <b>by </b><i>, , <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, , </i>;<br/><a NAME="E11:71_1_3"/><b>then </b><i><font color="Green">E456</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</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">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/><b>reconsider </b><font color="Maroon">domfc</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">fc</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E14:71_1_3"/><i><font color="Green">E457</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">domfc</font>
 <b>and </b><br/><a NAME="E15:71_1_3"/><i><font color="Green">E458</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">domfc</font> holds <br/><font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E16:71_1_3"/>
<font color="Maroon">domfc</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">domfc</font>
 ;<br/>

<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_3_1"/><i><font color="Green">E459</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> )
 ;<br/>
<a NAME="E4:71_1_3_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</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/>
<b>hence </b><a NAME="E5:71_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fc</font> = <font color="Maroon">fc</font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E18:71_1_3"/><i><font color="Green">E460</font></i>: 
<font color="Maroon">fc</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_3_2"/><i><font color="Green">E461</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fc</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_3_2"/><i><font color="Green">E462</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:71_1_3_2"/><i><font color="Green">E463</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:71_1_3_2"/><i><font color="Green">E464</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:71_1_3_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</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="E8:71_1_3_2"/><b>then </b><i><font color="Green">E465</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:71_1_3_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 
<b>by </b><i>, <a class="ref" href="graph_2.html#T5" target="_self">Th5</a>, , </i>;<br/>
<b>hence </b><a NAME="E10:71_1_3_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fc</font> = <font color="Maroon">fc</font> as    <a href="graph_2.html#M1" target="_self">FinSubsequence</a> of <font color="Maroon">c</font> <b>by </b><i><a class="txt" href="graph_2.html#E1:71"><i><font color="Green">E30</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">fc</font>
;<br/><b>set </b><font color="Maroon">domfvs</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">fvs</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E21:71_1_3"/><i><font color="Green">E466</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E22:71_1_3"/><i><font color="Green">E467</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  holds <br/><font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E23:71_1_3"/>
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_3_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_3_3"/><i><font color="Green">E468</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>
<a NAME="E4:71_1_3_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_3_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fvs</font> = <font color="Maroon">fvs</font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/><a NAME="E25:71_1_3"/><i><font color="Green">E469</font></i>: 
<font color="Maroon">fvs</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">vs</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_3_4"/><i><font color="Green">E470</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fvs</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_3_4"/><i><font color="Green">E471</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:71_1_3_4"/><i><font color="Green">E472</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:71_1_3_4"/><i><font color="Green">E473</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:71_1_3_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:71_1_3_4"/><b>then </b><i><font color="Green">E474</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:71_1_3_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kk</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E10:71_1_3_4"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">vs</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fvs</font> = <font color="Maroon">fvs</font> as    <a href="graph_2.html#M1" target="_self">FinSubsequence</a> of <font color="Maroon">vs</font> <b>by </b><i><a class="txt" href="graph_2.html#E1:71"><i><font color="Green">E30</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">fvs</font>
;<br/><b>take </b><font color="Maroon">c'</font> = <font color="Maroon">c1</font>;<br/><b>take </b><font color="Maroon">p1</font> = 1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>;<br/><a NAME="E27:71_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , , </i>;<br/><a NAME="E28:71_1_3"/><b>then </b><i><font color="Green">E475</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>hence </b><a NAME="E29:71_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E30:71_1_3"/>
<font color="Maroon">p1</font> <a href="graph_2.html#R1" target="_self">is_vertex_seq_of</a> <font color="Maroon">c'</font>
 <b>by </b><i>, , <a class="txt" href="graph_2.html#E4"><i><font color="Green">Lemma69</font></i></a>, , </i>;<br/><a NAME="E31:71_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="graph_2.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E32:71_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E33:71_1_3"/>
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E34:71_1_3"/>
<font color="Maroon">vs</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">vs</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</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">p1</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E35:71_1_3"/>
<font color="Maroon">domfc</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_3_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">domfc</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_3_5"/><i><font color="Green">E476</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> )
 ;<br/>
<a NAME="E4:71_1_3_5"/>
<font color="Maroon">k</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/>
<b>hence </b><a NAME="E5:71_1_3_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">domfc</font> = <font color="Maroon">domfc</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E37:71_1_3"/>
<font color="Maroon">domfc</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E38:71_1_3"/><b>then </b><i><font color="Green">E477</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font> <a href="hidden.html#R1">=</a>  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54">FINSEQ_3:54</a></i>;<br/><a NAME="E39:71_1_3"/><b>then </b><i><font color="Green">E478</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><b>set </b><font color="Maroon">SL</font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_3_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_3_6_1"/><i><font color="Green">E479</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_3_6_1"/><i><font color="Green">E480</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:71_1_3_6_1"/><i><font color="Green">E481</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:71_1_3_6_1"/><i><font color="Green">E482</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_3_6_1"/><b>then </b><i><font color="Green">E483</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:71_1_3_6_1"/><b>then </b><i><font color="Green">E484</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:71_1_3_6_1"/><i><font color="Green">E485</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">domfc</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E10:71_1_3_6_1"/><b>then </b><i><font color="Green">E486</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_3_6_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="graph_2.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_3_6_1"/><i><font color="Green">E487</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , , </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma90</font></i></a>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, , , , <a class="ref" href="graph_2.html#D1" target="_self">Def1</a>, , </i>
;<br/><b>hence </b><a NAME="E15:71_1_3_6_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:71_1_3_6"/><i><font color="Green">E488</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_3_6"/><i><font color="Green">E489</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:71_1_3_6"/><i><font color="Green">E490</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_3_6"/><b>then </b><i><font color="Green">E491</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fc</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E7:71_1_3_6"/><i><font color="Green">E492</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:71_1_3_6"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:71_1_3_6"/><i><font color="Green">E493</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> )
 ;<br/><a NAME="E11:71_1_3_6"/><i><font color="Green">E494</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fc</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:71_1_3_6"/><i><font color="Green">E495</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:71_1_3_6"/><i><font color="Green">E496</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:71_1_3_6"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:71_1_3_6"/><i><font color="Green">E497</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, </i>;<br/><font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , , , <a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="graph_2.html#D1" target="_self">Def1</a>, , , <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:71_1_3_6"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E41:71_1_3"/><b>then </b>
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fc</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfc</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E42:71_1_3"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E43:71_1_3"/>
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_3_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:71_1_3_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71_1_3_7"/><i><font color="Green">E498</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>
<a NAME="E4:71_1_3_7"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:71_1_3_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">domfvs</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E45:71_1_3"/>
<font color="Maroon">domfvs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><a NAME="E46:71_1_3"/><b>then </b><i><font color="Green">E499</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font> <a href="hidden.html#R1">=</a>  <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T54">FINSEQ_3:54</a></i>;<br/><a NAME="E47:71_1_3"/><b>then </b><i><font color="Green">E500</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/><b>set </b><font color="Maroon">SL</font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_3_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_3_8_1"/><i><font color="Green">E501</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> 1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:71_1_3_8_1"/><i><font color="Green">E502</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:71_1_3_8_1"/><i><font color="Green">E503</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default">1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#D1" target="_self">Def1</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:71_1_3_8_1"/><i><font color="Green">E504</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default">1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:71_1_3_8_1"/><b>then </b><i><font color="Green">E505</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:71_1_3_8_1"/><b>then </b><i><font color="Green">E506</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E9:71_1_3_8_1"/><i><font color="Green">E507</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">domfvs</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, , </i>;<br/><a NAME="E10:71_1_3_8_1"/><b>then </b><i><font color="Green">E508</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E11:71_1_3_8_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:71_1_3_8_1"/><i><font color="Green">E509</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, , , </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, , , , </i>
;<br/><b>hence </b><a NAME="E15:71_1_3_8_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:71_1_3_8"/><i><font color="Green">E510</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:71_1_3_8"/><i><font color="Green">E511</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E5:71_1_3_8"/><i><font color="Green">E512</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="graph_2.html#D1" target="_self">Def1</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:71_1_3_8"/><b>then </b><i><font color="Green">E513</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fvs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E7:71_1_3_8"/><i><font color="Green">E514</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E8:71_1_3_8"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">kk</font> where <font color="Olive">kk</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">kk</font> &amp; <font color="Olive">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:71_1_3_8"/><i><font color="Green">E515</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E11:71_1_3_8"/><i><font color="Green">E516</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fvs</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E12:71_1_3_8"/><i><font color="Green">E517</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span> <a href="recdef_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="txt" href="graph_2.html#E8"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E13:71_1_3_8"/><i><font color="Green">E518</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default">1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, , <a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:71_1_3_8"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="graph_2.html#T6" target="_self">Th6</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:71_1_3_8"/><i><font color="Green">E519</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, </i>;<br/><span class="p1">(<span class="default">1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="graph_2.html#T4" target="_self">Th4</a>, , <a class="ref" href="graph_2.html#T8" target="_self">Th8</a>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, , , <a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="graph_2.html#T7" target="_self">Th7</a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/><b>hence </b><a NAME="E18:71_1_3_8"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> 1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E49:71_1_3"/><b>then </b>
1,<font color="Maroon">n</font> <a href="graph_2.html#K2" target="_self">-cut</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fvs</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">domfvs</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E50:71_1_3"/>
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fvs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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