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<b>let </b><font color="Maroon">a</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">fs1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">fs2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>set </b><font color="Maroon">f</font> = <font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font>;<br/>
<a NAME="E1:52"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</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">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs1</font>,<font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <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"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs1</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs2</font>,<font color="Maroon">a</font></span>)</span></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">fs1</font></span>)</span> <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"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs2</font>,<font color="Maroon">a</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font>
 ;<br/><b>set </b><font color="Maroon">f1</font> =  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>,<font color="Maroon">a</font>;<br/><b>set </b><font color="Maroon">f2</font> = <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs1</font>,<font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font>;<br/><b>set </b><font color="Maroon">f3</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/><a NAME="E2:52_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</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">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:52_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</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">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:52"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:52_1_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs1</font> &amp; not <font color="Maroon">a</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">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b> <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>,<font color="Maroon">a</font> = 
<font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs1</font>,<font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font>
<b>by </b><i><a class="ref" href="wsierp_1.html#T5" target="_self">Th5</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:52_1_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
( <font color="Maroon">a</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">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs1</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:52"><i><font color="Green">E41</font></i></a>, <a class="ref" href="wsierp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">g1</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">g2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:52_1_1_1_2"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <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">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>
<br/>.= 
<font color="Maroon">a</font>

;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E7:52_1_1_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">fs1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:52"><i><font color="Green">E41</font></i></a>, <a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><a NAME="E8:52_1_1_1_2"/><i><font color="Green">E107</font></i>: 
 <a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs1</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T5" target="_self">Th5</a>, <a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, <a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a>, </i>;<br/><a NAME="E9:52_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, <a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E10:52_1_1_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><b>thus </b> <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>,<font color="Maroon">a</font> = 
<font color="Maroon">g1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g2</font>
<b>by </b><i><a class="ref" href="wsierp_1.html#T5" target="_self">Th5</a>, <a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs1</font>,<font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, <a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:52_1"/>
 <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs1</font>,<font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><b>set </b><font color="Maroon">f1</font> =  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></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">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span>;<br/><b>set </b><font color="Maroon">f2</font> = <font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs2</font>,<font color="Maroon">a</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs2</font> or <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_2_1_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs2</font>
 ;<br/><div class="add"><a NAME="E2:52_2_1_1_1"/><b>then </b><i><font color="Green">E112</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:52_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="wsierp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:52_2_1_1_1"/><b>then </b>
not <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</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">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b> <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></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">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span> = 
<font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs2</font>,<font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T5" target="_self">Th5</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_2_1_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs2</font>
 ;<br/><div class="add"><a NAME="E2:52_2_1_1_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs2</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:52"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:52_2_1_1_2"/><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</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">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="wsierp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:52_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:52"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E5:52_2_1_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</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">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">g1</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">g2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E7:52_2_1_1_2"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</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">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> <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">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E8:52_2_1_1_2"/><i><font color="Green">E124</font></i>: 
 <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></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">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a>, <a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, </i>;<br/><a NAME="E9:52_2_1_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E10:52_2_1_1_2"/>
<font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:52"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E11:52_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E13:52_2_1_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, , <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
;<br/><a NAME="E15:52_2_1_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">fs2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E16:52_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1</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">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E17:52_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E18:52_2_1_1_2"/><b>then </b>
 <a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs2</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T5" target="_self">Th5</a>, , </i>;<br/><b>hence </b><a NAME="E19:52_2_1_1_2"/>
 <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></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">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs2</font>,<font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:52_2"/>
 <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></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">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs2</font>,<font color="Maroon">a</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:52"/>
( ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs1</font> implies  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs1</font>,<font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font> ) &amp; ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 implies  <a href="finseq_3.html#K2">Del</a> <span class="p1">(<span class="default"><font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">fs2</font></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">fs1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fs1</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs2</font>,<font color="Maroon">a</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:52"><i><font color="Green">E40</font></i></a></i>;<br/>


</div>
