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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>







<b>set </b><font color="Maroon">x</font> = <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">q</font>;<br/>
<b>assume </b><a NAME="E1:6"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">p</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font>
 ;<br/>

<b>set </b><font color="Maroon">dx</font> =  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">dy</font> =  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">q</font>,<font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">xi</font> = <font color="Maroon">p</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">yi</font> = <font color="Maroon">q</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font>;<br/>
<a NAME="E2:6"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<a NAME="E3:6"/><i><font color="Green">E51</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> &amp;  <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">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> &amp;  <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">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:6"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><b>suppose </b></a><a NAME="E1:6_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:6_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><a NAME="E2:6_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><a NAME="E3:6_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:6_1_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E7:6_1_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E8:6_1_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T118">FINSEQ_3:118</a></i>;<br/><b>hence </b><a NAME="E9:6_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">q</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , , , , <a class="ref" href="finseq_3.html#T118">FINSEQ_3:118</a></i>;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E10:6_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">j</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> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 ;<br/><a NAME="E11:6_1_1_1"/><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_1_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T119">FINSEQ_3:119</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">q</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , , <a class="ref" href="finseq_3.html#T119">FINSEQ_3:119</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:6_1_1_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , , <a class="ref" href="finseq_3.html#T120">FINSEQ_3:120</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">q</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , , , , , <a class="ref" href="finseq_3.html#T120">FINSEQ_3:120</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:6_1_1"/>
 <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">q</font>,<font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2"><b>suppose </b></a><a NAME="E1:6_1_2"/>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:6_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q</font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i>, , <a class="ref" href="funct_7.html#D3">FUNCT_7:def 3</a></i>;<br/><b>hence </b><a NAME="E3:6_1_2"/>
 <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">q</font>,<font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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