<?xml version="1.0"?>
<div class="add">

<span class="kw">thus </span><a NAME="E1:20_1_1"/>
( <font color="Maroon" title="c2">s</font> <a href="compos_1.html#R1" title="COMPOS_1:pred.1">halts_at</a> <font color="Maroon" title="c3">l</font> implies <font color="Maroon" title="c2">s</font> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c1">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compos_1.html#D6" target="_self" title="COMPOS_1:def.6">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">s</font> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c1">S</font> implies <font color="Maroon" title="c2">s</font> <a href="compos_1.html#R1" title="COMPOS_1:pred.1">halts_at</a> <font color="Maroon" title="c3">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:20_1_1"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">s</font> <a href="compos_1.html#R1" title="COMPOS_1:pred.1">halts_at</a> <font color="Maroon" title="c3">l</font></span><br/>

<a NAME="E3:20_1_1"/>
<font color="Maroon" title="c3">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:20_1_1"/>
<font color="Maroon" title="c3">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="compos_1.html#D6" target="_self" title="COMPOS_1:def.6">COMPOS_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c1">S</font></span><br/>

<span class="kw">thus </span><a NAME="E5:20_1_1"/>
<font color="Maroon" title="c2">s</font> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
