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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being <a href="comput_1.html#V4" target="_self">homogeneous</a> <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> holds <br/> <a href="comput_1.html#K3" target="_self">arity</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Olive">g</font>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D6" target="_self">COMPUT_1:def 6</a><br/>

<b>assume </b><a NAME="E3:62"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font>, <font color="Maroon">g'</font> = <font color="Maroon">g</font> as    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E5:62"/><i><font color="Green">E52</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="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">g'</font>
 
<b>by </b><i>, </i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:62_1"/>
<font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E2:62_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g'</font> <a href="tarski.html#R1">c=</a> 0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:62_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g'</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">X</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>;<br/><a NAME="E4:62_1"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="relat_1.html#K1">dom</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="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><b>hence </b><a NAME="E5:62_1"/>
( <font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a> or  <a href="relat_1.html#K1">dom</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="relat_1.html#T64">RELAT_1:64</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E7:62"/>
( not <font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f'</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">X</font> ;<br/>
<b>take </b><font color="Maroon">n</font> =  <a href="comput_1.html#K3" target="_self">arity</a> <font color="Maroon">f'</font>;<br/>

<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E9:62"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/>


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