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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>assume </b><a NAME="E1:98"/><i><font color="Green" title="E61">A1</font></i>: 
<font color="Maroon" title="c2">e1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <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> , <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:98"/><i><font color="Green" title="E62">A2</font></i>: 
 <a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">G</font>
 <b>and </b><br/><a NAME="E4:98"/><i><font color="Green" title="E63">A3</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <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>  holds  <font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">p</font>
 <b>by </b><i><a class="ref" href="comput_1.html#D14" target="_self" title="COMPUT_1:def.14">Def14</a></i>;<br/>
<a NAME="E5:98"/><i><font color="Green" title="E64">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <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="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font>
 ;<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="E3:98_1_1"/><i><font color="Green" title="E65">A5</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> &amp; <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c8">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <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#E3:98_1_1"><i><font color="Green" title="E65">A5</font></i></a></i>;<br/><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">p</font> = 
 <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">p</font>
<b>by </b><i><a class="txt" href="comput_1.html#E4:98"><i><font color="Green" title="E63">A3</font></i></a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="comput_1.html#E1:98"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="comput_1.html#T57" target="_self" title="COMPUT_1:th.57">Th57</a></i>
;<br/><b>hence </b><a NAME="E6:98_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:98_1_1"><i><font color="Green" title="E65">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:98_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E3:98_1"/><b>then </b><i><font color="Green" title="E65">A6</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <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/><font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> = 
 <a href="comput_1.html#K8" title="COMPUT_1:func.8">primrec</a> <font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="txt" href="comput_1.html#E4:98"><i><font color="Green" title="E63">A3</font></i></a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="comput_1.html#E1:98"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="comput_1.html#T57" target="_self" title="COMPUT_1:th.57">Th57</a></i>
;<br/><b>hence </b><a NAME="E6:98_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E5:98"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="comput_1.html#E3:98_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:98"/><b>then </b><i><font color="Green" title="E65">A7</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">G</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="comput_1.html#E7:98"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>
;<br/>
<b>hence </b><a NAME="E9:98"/>
 <a href="comput_1.html#K9" title="COMPUT_1:func.9">primrec</a> <font color="Maroon" title="c2">e1</font>,<font color="Maroon" title="c3">e2</font>,<font color="Maroon" title="c1">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:98"><i><font color="Green" title="E62">A2</font></i></a></i>;<br/>


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