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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">f1</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="seqm_3.html#V6">to-naturals</a> <a href="comput_1.html#V2" target="_self">from-natural-fseqs</a> <a href="comput_1.html#V4" target="_self">homogeneous</a> <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">f2</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="seqm_3.html#V6">to-naturals</a> <a href="comput_1.html#V2" target="_self">from-natural-fseqs</a> <a href="comput_1.html#V4" target="_self">homogeneous</a> <a href="funct_1.html#NM1">Function</a>;<br/>





<b>assume </b><a NAME="E1:111"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">n</font> = <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">h</font> =  <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>;<br/>
<b>let </b><font color="Maroon">g</font> be    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">n'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:111"/><i><font color="Green">E49</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">n'</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n'</font> )
 <b>and </b><br/><a NAME="E3:111"/><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n'</font> &amp; <font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f2</font> )
 <b>and </b><br/><a NAME="E4:111"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n'</font> holds <br/>( ( <font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies  <a href="finseq_3.html#K2">Del</a> <font color="Olive">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> ) &amp; (  <a href="finseq_3.html#K2">Del</a> <font color="Olive">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> implies <font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) &amp; ( <font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Olive">p</font>,<font color="Maroon">i</font></span>)</span> ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( ( <font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies ( <font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> ) ) &amp; ( <font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> implies <font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) &amp; ( <font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">i</font>,<font color="Olive">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> ) ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D12" target="_self">COMPUT_1:def 12</a><br/>

<a NAME="E5:111"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( ( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> ) &amp; ( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> implies <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) &amp; ( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> ) );<br/><a NAME="E1:111_1"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E2:111_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:111_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( ( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> ) &amp; (  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> implies <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) &amp; ( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> implies  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> ) &amp; (  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> implies <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, <a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, , </i>;<br/><a NAME="E4:111_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies ( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">p</font>,<font color="Maroon">i</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, <a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, </i>;<br/><a NAME="E5:111_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">k</font> = <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>;<br/><a NAME="E6:111_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_7.html#T40">FUNCT_7:40</a></i>;<br/><a NAME="E7:111_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:111_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> iff <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:111_1_1"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> )
 ;<br/>

<a NAME="E3:111_1_1"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> iff ( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> ) )
 
<b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, <a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E4:111_1_1"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> iff ( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> ) )
 
<b>by </b><i><a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, </i>;<br/>
<b>thus </b><a NAME="E5:111_1_1"/>
( <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> iff <font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, <a class="ref" href="comput_1.html#T17" target="_self">Th17</a>, , </i>;<br/>

<b>assume </b><a NAME="E6:111_1_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E7:111_1_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, <a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="comput_1.html#T17" target="_self">Th17</a></i>;<br/>
<b>hence </b><a NAME="E8:111_1_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, <a class="ref" href="comput_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="comput_1.html#T14" target="_self">Th14</a>, <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, <a class="ref" href="comput_1.html#T18" target="_self">Th18</a>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E8:111_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="comput_1.html#T15" target="_self">Th15</a>, );<br/></i><b>hence </b><a NAME="E9:111_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> ) &amp; ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> ) &amp; ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:111"/><b>then </b><i><font color="Green">E142</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="comput_1.html#T12" target="_self">Th12</a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, , <a class="ref" href="subset_1.html#T8">SUBSET_1:8</a></i>;<br/>
<a NAME="E8:111"/><b>then </b>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:111"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
