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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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">E74</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E2:111"/><i><font color="Green">E75</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>and </b><br/><a NAME="E3:111"/><i><font color="Green">E76</font></i>: 
( <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></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">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E4:111"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> implies  <a href="finseq_3.html#K2">Del</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="finseq_3.html#K2">Del</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></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">c<sub>6</sub></font> implies ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>3</sub></font> ) ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>3</sub></font> implies <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></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">c<sub>6</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></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">c<sub>6</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_7.html#K3">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">E78</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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><a class="ref" href="comput_1.html#T60" target="_self">Th60</a></i>;<br/>
<div><i><font color="Green">E79</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">c<sub>8</sub></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">c<sub>2</sub></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="nat_1.html#NM1">Nat</a>] means ( ( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></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">c<sub>6</sub></font> implies <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> ) &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> implies <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></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">c<sub>6</sub></font> ) &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></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">c<sub>6</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> ) );<br/><a NAME="E1:111_1"/><i><font color="Green">E80</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></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">E81</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:111"><i><font color="Green">E74</font></i></a>, <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">E82</font></i>: 
( ( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> implies  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> implies  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="finseq_3.html#K2">Del</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="comput_1.html#E3:111"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E4:111"><i><font color="Green">E77</font></i></a>, <a class="txt" href="comput_1.html#E1:111_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E81"><i><font color="Green">Lemma70</font></i></a></i>;<br/><a NAME="E4:111_1"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> implies ( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="comput_1.html#E3:111"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E4:111"><i><font color="Green">E77</font></i></a>, <a class="txt" href="comput_1.html#E1:111_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E2:111_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E81"><i><font color="Green">Lemma70</font></i></a></i>;<br/><a NAME="E5:111_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="txt" href="comput_1.html#E3:111_1"><i><font color="Green">E82</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E6:111_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></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">E85</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:111_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> iff <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:111_1_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> )
 ;<br/>

<a NAME="E3:111_1_1"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font> iff ( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>3</sub></font> ) )
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:111"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E4:111"><i><font color="Green">E77</font></i></a>, <a class="txt" href="comput_1.html#E1:111_1"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E4:111_1_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> iff ( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>3</sub></font> ) )
 
<b>by </b><i><a class="txt" href="comput_1.html#E2:111_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E81"><i><font color="Green">Lemma70</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:111_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font> iff <font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="comput_1.html#E2:111_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E1:111_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="comput_1.html#E2:111_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="comput_1.html#E3:111_1_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="comput_1.html#E81"><i><font color="Green">Lemma70</font></i></a></i>;<br/>

<b>assume </b><a NAME="E6:111_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font>
 ;<br/>

<a NAME="E7:111_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:111"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E4:111"><i><font color="Green">E77</font></i></a>, <a class="txt" href="comput_1.html#E1:111_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E2:111_1_1"><i><font color="Green">E87</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:111_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="comput_1.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:111"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E4:111"><i><font color="Green">E77</font></i></a>, <a class="txt" href="comput_1.html#E1:111_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E2:111_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E1:111_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="comput_1.html#E4:111_1_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="comput_1.html#E6:111_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="comput_1.html#E81"><i><font color="Green">Lemma70</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:111_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="comput_1.html#E5:111_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E7:111_1"><i><font color="Green">E85</font></i></a>);<br/></i><b>hence </b><a NAME="E9:111_1"/>
( ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>8</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> ) &amp; ( <font color="Maroon">c<sub>8</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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> implies <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> ) )
 <b>by </b><i><a class="txt" href="comput_1.html#E6:111_1"><i><font color="Green">E84</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:111"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E2:111"><i><font color="Green">E75</font></i></a>, <a class="txt" href="comput_1.html#E3:111"><i><font color="Green">E76</font></i></a>, <a class="txt" href="comput_1.html#E5:111"><i><font color="Green">E78</font></i></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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E5:111"><i><font color="Green">E78</font></i></a>, <a class="txt" href="comput_1.html#E6:111"><i><font color="Green">E79</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:111"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="comput_1.html#E7:111"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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