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<b>let </b><font color="Maroon" title="c1">i</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> ;<br/><b>let </b><font color="Maroon" title="c2">e1</font>, <font color="Maroon" title="c3">e2</font> be  <a href="comput_1.html#NV2" title="COMPUT_1:NV.2">to-naturals</a> <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> <a href="comput_1.html#V2" title="COMPUT_1:attr.2">from-natural-fseqs</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>





<b>set </b><font color="Maroon" title="c4">f1</font> = <font color="Maroon" title="c2">e1</font>;<br/>
<b>set </b><font color="Maroon" title="c5">f2</font> = <font color="Maroon" title="c3">e2</font>;<br/>
<b>let </b><font color="Maroon" title="c6">g</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:113"/><i><font color="Green" title="E77">A1</font></i>: 
<font color="Maroon" title="c2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>assume </b><a NAME="E2:113"/>
<font color="Maroon" title="c6">g</font> <a href="comput_1.html#R1" title="COMPUT_1:pred.1">is_primitive-recursively_expressed_by</a> <font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">n</font> being    <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> <b> such that </b><br/><a NAME="E4:113"/><i><font color="Green" title="E78">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">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>and </b><br/><a NAME="E5:113"/>
( <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &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="c7">n</font> )
 <b>and </b><br/><a NAME="E6:113"/>
( <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c2">e1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">e2</font> )
 <b>and </b><br/><a NAME="E7:113"/><i><font color="Green" title="E79">A3</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> holds <br/>( ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> implies  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">e1</font> ) &amp; (  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">e1</font> implies <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> ) &amp; ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> implies <font color="Maroon" title="c6">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( for <font color="Olive" title="b2">n</font> being   <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>  holds <br/> ( ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> implies ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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="c6">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">e2</font> ) ) &amp; ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</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="c6">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">e2</font> implies <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> ) &amp; ( <font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> implies <font color="Maroon" title="c6">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <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="c6">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_7.html#K3" title="FUNCT_7:func.3">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Olive" title="b2">n</font></span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span> ) ) ) )
 <b>by </b><i><a class="ref" href="comput_1.html#D12" target="_self" title="COMPUT_1:def.12">Def12</a></i>;<br/>
<div><i><font color="Green" title="E80">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">y</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c7">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/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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> ] <b>means </b>not <font color="Maroon" title="c8">y</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font>;<br/><a NAME="E1:113_1"/><i><font color="Green" title="E81">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="E2:113_1"/><b>then </b><i><font color="Green" title="E82">A6</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="comput_1.html#E1:113"><i><font color="Green" title="E77">A1</font></i></a>, <a class="txt" href="comput_1.html#E7:113"><i><font color="Green" title="E79">A3</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/><a NAME="E3:113_1"/><i><font color="Green" title="E83">A7</font></i>: 
for <font color="Olive" title="b1">k</font> being   <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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="comput_1.html#E7:113"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="comput_1.html#E1:113_1"><i><font color="Green" title="E81">A5</font></i></a></i>;<br/><b>thus </b><a NAME="E4:113_1"/>
for <font color="Olive" title="b1">k</font> being   <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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="comput_1.html#E2:113_1"><i><font color="Green" title="E82">A6</font></i></a>, <a class="txt" href="comput_1.html#E3:113_1"><i><font color="Green" title="E83">A7</font></i></a>);</i><br/></div><b>end;</b></div>
<b>assume </b><a NAME="E9:113"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E10:113"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:113"/><i><font color="Green" title="E81">A8</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c7">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#E4:113"><i><font color="Green" title="E78">A2</font></i></a>, <a class="txt" href="comput_1.html#E12:113"><i><font color="Green" title="E81">A8</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">xi</font> = <font color="Maroon" title="c9">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>;<br/>
<a NAME="E14:113"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="comput_1.html#K1" title="COMPUT_1:func.1">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T37" title="FUNCT_7:th.37">FUNCT_7:37</a></i>;<br/>
<b>hence </b><a NAME="E15:113"/>
contradiction
 <b>by </b><i><a class="txt" href="comput_1.html#E8:113"><i><font color="Green" title="E80">A4</font></i></a>, <a class="txt" href="comput_1.html#E12:113"><i><font color="Green" title="E81">A8</font></i></a></i>;<br/>


</div>
