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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="E1:82"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c1">Gs</font> is <a href="lexbfs.html#V10" title="LEXBFS:attr.10">eventually-constant</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">n</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:82"/><i><font color="Green" title="E34">A2</font></i>: 
for <font color="Olive" title="b1">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c1">Gs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font> <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="Olive" title="b1">m</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E1:82"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="lexbfs.html#D16" target="_self" title="LEXBFS:def.16">Def16</a></i>;<br/>
<a NAME="E4:82"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:82"/><b>then </b><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c1">Gs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">n</font> <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"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E3:82"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>
<a NAME="E6:82"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E7:82"/>
<font color="Maroon" title="c1">Gs</font> is <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a>
 <b>by </b><i><a class="txt" href="lexbfs.html#E5:82"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="glib_000.html#D56" title="GLIB_000:def.56">GLIB_000:def 56</a></i>;<br/>


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