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

<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font></span><br/><b>let </b><font color="Maroon" title="c3">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="comput_1.html#V7" title="COMPUT_1:attr.7">primitive-recursively_closed</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )</span><br/>





<b>assume </b><a NAME="E1:116"/><i><font color="Green" title="E80">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:116_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#D10" target="_self" title="COMPUT_1:def.10">Def10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; ( 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> <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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"><span class="p2">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/><a NAME="E2:116_1"/><i><font color="Green" title="E81">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T40" target="_self" title="COMPUT_1:th.40">Th40</a></i>;<br/><a NAME="E3:116_1"/><b>then </b><i><font color="Green" title="E82">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_6.html#T61" title="FUNCT_6:th.61">FUNCT_6:61</a></i>;<br/><a NAME="E4:116_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E1:116"><i><font color="Green" title="E80">A1</font></i></a>, <a class="ref" href="comput_1.html#T40" target="_self" title="COMPUT_1:th.40">Th40</a></i>;<br/><a NAME="E5:116_1"/><b>then </b><i><font color="Green" title="E83">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self" title="COMPUT_1:th.11">Th11</a></i>;<br/><a NAME="E6:116_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#D10" target="_self" title="COMPUT_1:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E7:116_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E3:116_1"><i><font color="Green" title="E82">A3</font></i></a>, <a class="txt" href="comput_1.html#E5:116_1"><i><font color="Green" title="E83">A4</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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> <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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"><span class="p2">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><b>assume </b><a NAME="E8:116_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E10:116_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E11:116_1"/><b>then </b><i><font color="Green" title="E84">A5</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">x'</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:116"><i><font color="Green" title="E80">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>set </b><font color="Maroon" title="c6">xi</font> = <font color="Maroon" title="c5">x'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>;<br/><i><font color="Green" title="E85">A6</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="comput_1.html#D10" target="_self" title="COMPUT_1:def.10">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="comput_1.html#E11:116_1"><i><font color="Green" title="E84">A5</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x'</font> = 
<span class="p1">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x'</font></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E3:116_1"><i><font color="Green" title="E82">A3</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x'</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="txt" href="comput_1.html#E2:116_1"><i><font color="Green" title="E81">A2</font></i></a>, <a class="ref" href="funct_6.html#T61" title="FUNCT_6:th.61">FUNCT_6:61</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="comput_1.html#T42" target="_self" title="COMPUT_1:th.42">Th42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">x'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="comput_1.html#D10" target="_self" title="COMPUT_1:def.10">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x'</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
;<br/><b>hence </b><a NAME="E14:116_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p3"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E12:116_1"><i><font color="Green" title="E85">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:116"/><b>then </b><i><font color="Green" title="E81">A7</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E4:116"/><i><font color="Green" title="E82">A8</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="comput_1.html#V5" title="COMPUT_1:attr.5">composition_closed</a>
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/>
<a NAME="E5:116"/><i><font color="Green" title="E83">A9</font></i>: 
1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>;<br/>
<i><font color="Green" title="E84">A10</font></i>:  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <span class="p1">(<span class="default">1 <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> 1</span>)</span> = 
1
<b>by </b><i><a class="ref" href="comput_1.html#T38" target="_self" title="COMPUT_1:th.38">Th38</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E7:116"/><i><font color="Green" title="E85">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )</span><br/>

<b>assume </b><a NAME="E1:116_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font></span><br/>

<a NAME="E2:116_3"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E3:116_3"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:116_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:116"><i><font color="Green" title="E80">A1</font></i></a>, <a class="ref" href="comput_1.html#D17" target="_self" title="COMPUT_1:def.17">Def17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c4">nproji</font> = <font color="Maroon" title="c2">n</font> <a href="comput_1.html#K6" title="COMPUT_1:func.6">proj</a> <font color="Maroon" title="c1">i</font> as    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="comput_1.html#T31" target="_self" title="COMPUT_1:th.31">Th31</a></i>;<br/>
<a NAME="E9:116"/>
<span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">nproji</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> is  <a href="comput_1.html#V4" title="COMPUT_1:attr.4">with_the_same_arity</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="comput_1.html#K3" title="COMPUT_1:func.3">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<b>hence </b><a NAME="E10:116"/>
<font color="Maroon" title="c2">n</font> <a href="comput_1.html#K5" title="COMPUT_1:func.5">succ</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:116"><i><font color="Green" title="E81">A7</font></i></a>, <a class="txt" href="comput_1.html#E4:116"><i><font color="Green" title="E82">A8</font></i></a>, <a class="txt" href="comput_1.html#E5:116"><i><font color="Green" title="E83">A9</font></i></a>, <a class="txt" href="comput_1.html#E6:116"><i><font color="Green" title="E84">A10</font></i></a>, <a class="txt" href="comput_1.html#E7:116"><i><font color="Green" title="E85">A11</font></i></a>, <a class="ref" href="comput_1.html#D15" target="_self" title="COMPUT_1:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
