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

<b>thus </b><a NAME="E1:47_1_1"/>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font> implies ( <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<font color="Maroon">a</font> iff <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font> ) )
 <b>by </b><i><a class="ref" href="finseq_3.html#T113">FINSEQ_3:113</a></i>;<br/>

<b>assume </b><a NAME="E2:47_1_1"/><i><font color="Green">E40</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">fs</font>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:47_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<font color="Maroon">a</font>
 ;<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="E3:47_1_1_1"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T113">FINSEQ_3:113</a></i>;<br/><b>thus </b><a NAME="E4:47_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><a NAME="E5:47_1_1_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> implies <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T119">FINSEQ_3:119</a></i>;<br/><b>assume </b><a NAME="E6:47_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_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="E1:47_1_1_1_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> or <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:47_1_1_1_1_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_1_1_1_1"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:47_1_1_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font>
 ;<br/><div class="add"><a NAME="E2:47_1_1_1_1_2"/><b>then </b>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:47_1_1_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E4:47_1_1_1_1_2"/>
<font color="Maroon">b</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:47_1_1_1_1_2"/><b>then </b>
not <font color="Maroon">b</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E6:47_1_1_1_1_2"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:47_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:47_1_1"/><i><font color="Green">E105</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> &amp; ( for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( ( <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> implies <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> ) &amp; ( <font color="Olive">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> implies <font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) ) ) )
 ;<br/>

<b>consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:47_1_1"/><i><font color="Green">E106</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp;  <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">fs</font>,<font color="Maroon">a</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#T113">FINSEQ_3:113</a></i>;<br/>
<a NAME="E7:47_1_1"/><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <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">fs</font>,<font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:47_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 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">IT</font> holds <br/><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:47_1_1_2"/><i><font color="Green">E108</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">IT</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1_2_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:47_1_1_2_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:47_1_1_2_1_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:47_1_1_2_1_1"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="wsierp_1.html#E2:47_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_3.html#T119">FINSEQ_3:119</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:47_1_1_2_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:47_1_1_2_1_2"/><b>then </b>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:47_1_1_2_1_2"/>
<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="wsierp_1.html#T2" target="_self">Th2</a>, <a class="txt" href="wsierp_1.html#E2:47_1_1"><i><font color="Green">E40</font></i></a>, <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="E9:47_1_1"/>
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


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