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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">e1</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>let </b><font color="Maroon">e2</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">f1</font> = <font color="Maroon">e1</font>;<br/>
<b>set </b><font color="Maroon">f2</font> = <font color="Maroon">e2</font>;<br/>
<b>assume </b><a NAME="E1:100"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">e1</font> is <a href="xboole_0.html#V1">empty</a>
 ;<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">e1</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="E3:100"/><i><font color="Green">E49</font></i>: 
 <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</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="E4:100"/><i><font color="Green">E51</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">e1</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">e1</font>,<font color="Maroon">e2</font>,<font color="Maroon">i</font>,<font color="Olive">p</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:100"/><i><font color="Green">E52</font></i>: 
 <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">e1</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:100_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:100_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</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">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">p</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">e1</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#T8" target="_self">Th8</a></i>;<br/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> = 
 <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font>,<font color="Maroon">i</font>,<font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E6:100_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="comput_1.html#T8" target="_self">Th8</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:100_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E3:100_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon">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">e1</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/><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> = 
 <a href="comput_1.html#K9" target="_self">primrec</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font>,<font color="Maroon">i</font>,<font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E6:100_1"/>
<font color="Maroon">y</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="comput_1.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:100"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
 <a href="card_3.html#K3">Union</a> <font color="Maroon">G</font> = 
 <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/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="comput_1.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
;<br/>
<b>hence </b><a NAME="E9:100"/>
 <a href="comput_1.html#K10" target="_self">primrec</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font>,<font color="Maroon">i</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/>


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