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<b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="comput_1.html#V8" target="_self">primitive-recursively_closed</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> </span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="comput_1.html#M1" target="_self">Element</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">F</font> be  <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">P</font>;<br/>





<b>assume </b><a NAME="E1:121"/><i><font color="Green">E48</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> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:121"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:121_1"/>
( <font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">f</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="121_1_1"><b>suppose </b></a><a NAME="E1:121_1_1"/>
<font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:121_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">F</font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T62">RELAT_1:62</a></i>;<br/><b>hence </b><a NAME="E3:121_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">F</font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="121_1_2"><b>suppose </b></a><a NAME="E1:121_1_2"/>
not <font color="Maroon">f</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><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> <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:121_1_2"/>
( <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="finseq_2.html#K3">*</a>  &amp; <font color="Maroon">P</font> <a href="finseq_2.html#K3">*</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="finseq_2.html#K3">*</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a>, <a class="ref" href="finseq_1.html#T83">FINSEQ_1:83</a></i>;<br/><b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> as  <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>  <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="finseq_2.html#K3">*</a>  ;<br/><a NAME="E5:121_1_2"/>
<font color="Maroon">P</font> is <a href="comput_1.html#V6" target="_self">composition_closed</a>
 <b>by </b><i><a class="ref" href="comput_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E6:121_1_2"/><b>then </b>
<font color="Maroon">f'</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">F'</font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="comput_1.html#T16" target="_self">Th16</a>, </i>;<br/><b>hence </b><a NAME="E7:121_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">F</font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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