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<div class="add">

<b>let </b><font color="Maroon">Gs</font> be  <a href="lexbfs.html#V9" target="_self">iterative</a> <a href="lexbfs.html#V10" target="_self">eventually-constant</a>  <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">GL</font> = <font color="Maroon">Gs</font> <a href="glib_000.html#K40">.Lifespan()</a> ;<br/>
<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:84"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">Gs</font> <a href="glib_000.html#K40">.Lifespan()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">Gs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Gs</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Gs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Gs</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E2:84"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:84"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:84_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:84_1"/>
<font color="Maroon">Gs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Gs</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Gs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Gs</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:84_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i><a class="ref" href="glib_000.html#D57">GLIB_000:def 57</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:84"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, <a class="txt" href="lexbfs.html#E2:84"><i><font color="Green">E36</font></i></a>);<br/></i>
<a NAME="E5:84"/>
 ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1">(<span class="default"><font color="Maroon">Gs</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E1:84"><i><font color="Green">E34</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>hence </b><a NAME="E6:84"/>
<font color="Maroon">Gs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Gs</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span>
 <b>by </b><i/>;<br/>


</div>
