<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">b</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">D</font>, <font color="Maroon">G</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">D</font>, <font color="Maroon">bFG</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:75"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">b</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">b</font> is <a href="binop_1.html#V2">associative</a> )
 <b>and </b><br/><a NAME="E2:75"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">b</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  or  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>and </b><br/><a NAME="E3:75"/><i><font color="Green">E51</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">G</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">bFG</font> )
 <b>and </b><br/><a NAME="E4:75"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">bFG</font> holds <br/><font color="Maroon">bFG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">xF</font> =  <a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">xG</font> =  <a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">xb</font> =  <a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font>;<br/>
<a NAME="E5:75"/><i><font color="Green">E57</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">F</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">G</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">bFG</font> )
 
<b>by </b><i><a class="ref" href="prgcor_2.html#D2">PRGCOR_2:def 2</a></i>;<br/>
<a NAME="E6:75"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><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="E1:75_1"/><i><font color="Green">E58</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"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span>
 ;<br/>

<a NAME="E2:75_1"/>
<font color="Maroon">k</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="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:75_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="binop_2.html#K10">-</a> 1 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="nat_1.html#T61">NAT_1:61</a></i>;<br/>
<a NAME="E5:75_1"/>
<font color="Maroon">k</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="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:75_1"/><b>then </b>
( <font color="Maroon">k1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k1</font> <a href="binop_2.html#K23">+</a> 1 &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"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:75_1"/><b>then </b>
<font color="Maroon">k1</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"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:75_1"/><b>then </b>
( <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">bFG</font> &amp; <font color="Maroon">k</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="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="card_fin.html#T3" target="_self">Th3</a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:75_1"/><b>then </b>
( <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">bFG</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>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">bFG</font> )
 
<b>by </b><i>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:75_1"/><b>then </b>
( <font color="Maroon">bFG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font></span>)</span> &amp; <font color="Maroon">bFG</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="prgcor_2.html#D2">PRGCOR_2:def 2</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E11:75_1"/>
<span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:75"/><b>then </b>
( <font color="Maroon">b</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">bFG</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finsop_1.html#K2">"**"</a> <span class="p2">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">F</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finsop_1.html#K2">"**"</a> <span class="p2">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">G</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="finsop_1.html#K2">"**"</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K2">XFS2FS</a> <font color="Maroon">G</font></span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="finsop_1.html#T10">FINSOP_1:10</a>, <a class="txt" href="card_fin.html#E2"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E8:75"/><b>then </b>
<font color="Maroon">b</font> <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">bFG</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="card_fin.html#E2"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:75"/>
<font color="Maroon">b</font> <a href="stirl2_1.html#K4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="afinsq_1.html#K5">^</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">bFG</font>
 <b>by </b><i>, , , <a class="ref" href="stirl2_1.html#T47">STIRL2_1:47</a></i>;<br/>


</div>
