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<span class="kw">let </span><font color="Maroon" title="c1">Gs</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">Gs</font> is  <a href="lexbfs.html#V4" title="LEXBFS:attr.4">eventually-constant</a>  implies <font color="Maroon" title="c1">Gs</font> is  <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:30"/>
<font color="Maroon" title="c1">Gs</font> is  <a href="lexbfs.html#V4" title="LEXBFS:attr.4">eventually-constant</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">Gs</font> is  <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:30"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lexbfs.html#D7" target="_self" title="LEXBFS:def.7">Def7</a></span>;<br/>
<a NAME="E4:30"/>
<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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:30"/><span class="kw">then </span>
<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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:30"/>
<font color="Maroon" title="c1">Gs</font> is  <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D56" title="GLIB_000:def.56">GLIB_000:def 56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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