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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><b>that </b><br/><a NAME="E1:102"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f1</font> is <a href="comput_1.html#V3" target="_self">len-total</a>
 <b>and </b><br/><a NAME="E2:102"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f2</font> is <a href="comput_1.html#V3" target="_self">len-total</a>
 <b>and </b><br/><a NAME="E3:102"/><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> 2 <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="E4:102"/><i><font color="Green">E52</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f1</font></span>)</span> )
 ;<br/>

<a NAME="E5:102"/><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/>
<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/>
<a NAME="E6:102"/>
<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>  <a href="tarski.html#R1">c=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:102_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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> 
 ;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <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>  <b>by </b><i><a class="ref" href="comput_1.html#T10" target="_self">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</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> , <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:102_1"/><i><font color="Green">E55</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K3">Union</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E5:102_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">p</font> being   <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>  holds  <font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>,<font color="Olive">p</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:102_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>,<font color="Maroon">x'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:102_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">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#K9" target="_self">primrec</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>,<font color="Maroon">i</font>,<font color="Maroon">x'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E12:102_1"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="comput_1.html#T8" target="_self">Th8</a>, , </i>;<br/>
<a NAME="E8:102_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</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><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:102_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E10:102_1"/>
 <a href="card_3.html#K3">Union</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">rngG</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a> <a href="comput_1.html#V1" target="_self">compatible</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, <a class="ref" href="comput_1.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E12:102_1"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">rngG</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngG</font> : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E13:102_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">f</font></span>)</span> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">rngG</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E14:102_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">rngG</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#D1" target="_self">Def1</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E15:102_1"/>
<font color="Maroon">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>
 <b>by </b><i><a class="ref" href="comput_1.html#T11" target="_self">Th11</a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:102"/>
 <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="hidden.html#R1">=</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>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>

<b>hence </b><a NAME="E8:102"/>
 <a href="comput_1.html#K3" target="_self">arity</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="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/>;<br/>


</div>
