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

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="compos_1.html#V3" title="COMPOS_1:attr.3">standard-ins</a>   <a href="compos_1.html#V6" title="COMPOS_1:attr.6">homogeneous</a>   <a href="compos_1.html#V7" title="COMPOS_1:attr.7">regular</a>   <a href="compos_1.html#V8" title="COMPOS_1:attr.8">J/A-independent</a>   <a href="compos_1.html#V10" title="COMPOS_1:attr.10">proper-halt</a>   <a href="compos_1.html#L1" title="COMPOS_1:struct.1">COM-Struct</a>  holds   <a href="compos_1.html#K10" title="COMPOS_1:func.10">IncAddr</a> (<span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Olive" title="b1">S</font></span>)</span>,<font color="Maroon" title="c1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Olive" title="b1">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="compos_1.html#V3" title="COMPOS_1:attr.3">standard-ins</a>   <a href="compos_1.html#V6" title="COMPOS_1:attr.6">homogeneous</a>   <a href="compos_1.html#V7" title="COMPOS_1:attr.7">regular</a>   <a href="compos_1.html#V8" title="COMPOS_1:attr.8">J/A-independent</a>   <a href="compos_1.html#V10" title="COMPOS_1:attr.10">proper-halt</a>   <a href="compos_1.html#L1" title="COMPOS_1:struct.1">COM-Struct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="compos_1.html#K10" title="COMPOS_1:func.10">IncAddr</a> (<span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span><br/>



<span class="lab"><font color="Green" title="E40">A1</font></span>:  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">IncAddr</a> (<span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span> = 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="compos_1.html#D19" target="_self" title="COMPOS_1:def.19">Def40</a></span>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="compos_1.html#E29"><span class="lab"><font color="Green" title="E20">Lm12</font></span></a></span>
;<br/>
<a NAME="E2:100"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="compos_1.html#E29"><span class="lab"><font color="Green" title="E20">Lm12</font></span></a></span>;<br/>
<a NAME="E3:100"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><span class="p1">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">IncAddr</a> (<span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <span class="p1">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">IncAddr</a> (<span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:100_2"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">IncAddr</a> (<span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/>

<a NAME="E2:100_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E44">A5</font></span>: <span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
<span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E1:100_2"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c2">S</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>
;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="compos_1.html#K10" title="COMPOS_1:func.10">IncAddr</a> (<span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c1">k</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
 <a href="compos_1.html#K9" title="COMPOS_1:func.9">IncAddr</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c1">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E1:100_2"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E2:100_2"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="compos_1.html#D19" target="_self" title="COMPOS_1:def.19">Def40</a></span>
<br/>.= 
 <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c2">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100_2"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="ref" href="compos_1.html#T11" target="_self" title="COMPOS_1:th.11">Th92</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100_2"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:100"/>
 <a href="compos_1.html#K10" title="COMPOS_1:func.10">IncAddr</a> (<span class="p1">(<span class="default"><a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font></span>)</span>,<font color="Maroon" title="c1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</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>
