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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  <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>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</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>assume </b><a NAME="E1:116"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>assume </b><a NAME="E2:116"/>
<font color="Maroon">c<sub>6</sub></font> <a href="comput_1.html#R1" target="_self">is_primitive-recursively_expressed_by</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>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:116"/><i><font color="Green">E81</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> 
 <b>and </b><br/><a NAME="E5:116"/>
( <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="E6:116"/>
( <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="E7:116"/><i><font color="Green">E82</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> ) ) ) )
 <b>by </b><i><a class="ref" href="comput_1.html#D12" target="_self">Def12</a></i>;<br/>
<div><i><font color="Green">E83</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="116_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 <font color="Maroon">c<sub>7</sub></font> <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 not <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>;<br/><a NAME="E1:116_1"/><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E2:116_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="txt" href="comput_1.html#E1:116"><i><font color="Green">E80</font></i></a>, <a class="txt" href="comput_1.html#E7:116"><i><font color="Green">E82</font></i></a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><a NAME="E3:116_1"/><i><font color="Green">E86</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]
 <b>by </b><i><a class="txt" href="comput_1.html#E7:116"><i><font color="Green">E82</font></i></a>, <a class="txt" href="comput_1.html#E1:116_1"><i><font color="Green">E84</font></i></a></i>;<br/><b>thus </b><a NAME="E4:116_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#E2:116_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="comput_1.html#E3:116_1"><i><font color="Green">E86</font></i></a>);</i><br/></div><b>end;</b></div>
<b>assume </b><a NAME="E9:116"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E10:116"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:116"/><i><font color="Green">E84</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>7</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="comput_1.html#E4:116"><i><font color="Green">E81</font></i></a>, <a class="txt" href="comput_1.html#E12:116"><i><font color="Green">E84</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E14:116"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</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>9</sub></font> <a href="comput_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T37">FUNCT_7:37</a></i>;<br/>
<b>hence </b><a NAME="E15:116"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E8:116"><i><font color="Green">E83</font></i></a>, <a class="txt" href="comput_1.html#E12:116"><i><font color="Green">E84</font></i></a></i>;<br/>


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