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<b>let </b><font color="Maroon" title="c1">Gs</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:80"/><i><font color="Green" title="E32">A1</font></i>: 
( <font color="Maroon" title="c1">Gs</font> is <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a> &amp; <font color="Maroon" title="c1">Gs</font> is <a href="lexbfs.html#V9" title="LEXBFS:attr.9">iterative</a> )
 ;<br/>

<b>set </b><font color="Maroon" title="c2">GL</font> = <font color="Maroon" title="c1">Gs</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</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="c1">Gs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Gs</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Gs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">Gs</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/>
<a NAME="E2:80"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E3:80"/><i><font color="Green" title="E34">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:80_1"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/>

<a NAME="E2:80_1"/>
<font color="Maroon" title="c1">Gs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">Gs</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Gs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">Gs</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E1:80"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="lexbfs.html#E1:80_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="lexbfs.html#D15" target="_self" title="LEXBFS:def.15">Def15</a></i>;<br/>
<b>hence </b><a NAME="E3:80_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:80"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="glib_000.html#D57" title="GLIB_000:def.57">GLIB_000:def 57</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:80"/><i><font color="Green" title="E35">A5</font></i>: 
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="lexbfs.html#E2:80"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="lexbfs.html#E3:80"><i><font color="Green" title="E34">A3</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:80_2"/><i><font color="Green" title="E36">A6</font></i>: 
<font color="Maroon" title="c1">Gs</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 ;<br/><a NAME="E2:80_2"/>
 ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <span class="p1">(<span class="default"><font color="Maroon" title="c1">Gs</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:80_2"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>hence </b><a NAME="E3:80_2"/>
<font color="Maroon" title="c1">Gs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Gs</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Gs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E4:80"><i><font color="Green" title="E35">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:80"/>
<font color="Maroon" title="c1">Gs</font> is <a href="lexbfs.html#V10" title="LEXBFS:attr.10">eventually-constant</a>
 <b>by </b><i><a class="ref" href="lexbfs.html#D16" target="_self" title="LEXBFS:def.16">Def16</a></i>;<br/>


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