<?xml version="1.0"?>
<div class="add">

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

<b>set </b><font color="Maroon">X</font> = <font color="Maroon">D</font>;<br/>
<b>let </b><font color="Maroon">f</font> be  <a href="unialg_1.html#V2">quasi_total</a>  <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</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  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:88"/><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>
 <b>and </b><br/><a NAME="E2:88"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">h</font> being   <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font>  st <font color="Olive">h</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> holds <br/><font color="Olive">h</font> is <a href="unialg_1.html#V2">quasi_total</a>
 ;<br/>

<b>set </b><font color="Maroon">fF</font> = <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>;<br/>
<b>reconsider </b><font color="Maroon">g</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">D</font> <b>by </b><i><a class="ref" href="partfun1.html#T56">PARTFUN1:56</a></i>;<br/>
<a NAME="E4:88"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<a NAME="E5:88"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">D</font> : <font color="Olive">h</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span> 
 
;<br/>
<a NAME="E6:88"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">g</font> is <a href="unialg_1.html#V2">quasi_total</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>, <font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D2">UNIALG_1:def 2</a><br/>

<b>assume </b><a NAME="E1:88_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> &amp; <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> )
 ;<br/>

<b>hence </b><a NAME="E2:88_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:88"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">g</font> is  <a href="unialg_1.html#V2">quasi_total</a>  <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:88_2"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  ex <font color="Olive">h</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">h</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp; <font color="Olive">h</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">h</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">h</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> holds <br/><font color="Olive">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_2_1"><b>suppose </b></a><a NAME="E1:88_2_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:88_2_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> is  <a href="unialg_1.html#V2">quasi_total</a>  <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T62">RELAT_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_2_2"><b>suppose </b></a><a NAME="E1:88_2_2"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:88_2_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#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_6.html#T60">FUNCT_6:60</a></i>;<br/><b>hence </b><a NAME="E3:88_2_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> is  <a href="unialg_1.html#V2">quasi_total</a>  <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_2_3"><b>suppose </b></a><a NAME="E1:88_2_3"/>
 ex <font color="Olive">h</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">h</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp; <font color="Olive">h</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:88_2_3"/><b>then </b>
<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/>;<br/><b>hence </b><a NAME="E3:88_2_3"/>
<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> is  <a href="unialg_1.html#V2">quasi_total</a>  <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T6" target="_self">Th6</a>, <a class="ref" href="relat_1.html#T62">RELAT_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_2_4"><b>suppose </b></a><a NAME="E1:88_2_4"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">h</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">h</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> holds <br/><font color="Olive">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 ;<br/><div class="add"><a NAME="E2:88_2_4"/><b>then </b>
for <font color="Olive">h</font> being   <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font>  st <font color="Olive">h</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> holds <br/>( <font color="Olive">h</font> is <a href="unialg_1.html#V2">quasi_total</a> &amp; not <font color="Olive">h</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="txt" href="comput_1.html#E7:88"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E3:88_2_4"/>
<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> is  <a href="unialg_1.html#V2">quasi_total</a>  <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E6:88"><i><font color="Green">E52</font></i></a>, , <a class="ref" href="comput_1.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
