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<b>let </b><font color="Maroon" title="c1">E</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K10" title="AFINSQ_1:func.10">^omega</a> </span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> $1</span>)</span>;<br/>
<a NAME="E1:23"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>thus </b><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1"><a href="flang_1.html#K4" title="FLANG_1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span>
<b>by </b><i><a class="ref" href="flang_1.html#T14" title="FLANG_1:th.14">FLANG_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="flang_1.html#T25" title="FLANG_1:th.25">FLANG_1:25</a></i>
;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E20">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:23_2"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ;<br/><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="txt" href="flang_3.html#E1:23_2"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="flang_3.html#T16" target="_self" title="FLANG_3:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">A</font></span>)</span>
<b>by </b><i><a class="ref" href="flang_1.html#T19" title="FLANG_1:th.19">FLANG_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="flang_1.html#T24" title="FLANG_1:th.24">FLANG_1:24</a></i>
;<br/><b>hence </b><a NAME="E3:23_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:23"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="flang_3.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="flang_3.html#E2:23"><i><font color="Green" title="E20">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:23"/>
<font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c4">k</font></span>)</span>
 ;<br/>


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