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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be  <a href="comput_1.html#V4" target="_self">homogeneous</a> <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">Y</font>;<br/><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="E1:43"/><i><font color="Green">E48</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">f</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>suppose </b></a><a NAME="E1:43_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:43_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T19">FUNCT_7:19</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:43_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:43_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="finseq_2.html#K3">*</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E5:43_1_1"/>
0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</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/>;<br/><b>hence </b><a NAME="E6:43_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_7.html#T19">FUNCT_7:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2"><b>suppose </b></a><a NAME="E1:43_1_2"/>
not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X'</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E4:43_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:43_1_2"/><b>then </b>
<font color="Maroon">x'</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X'</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/><b>hence </b><a NAME="E6:43_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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