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

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">X</font> = <font color="Maroon" title="c1">D</font>;<br/>
<b>let </b><font color="Maroon" title="c3">f</font> be  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be  <a href="comput_1.html#V4" title="COMPUT_1:attr.4">with_the_same_arity</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:86"/><i><font color="Green" title="E50">A1</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E2:86"/><i><font color="Green" title="E51">A2</font></i>: 
for <font color="Olive" title="b1">h</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">h</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="c4">F</font> holds <br/><font color="Olive" title="b1">h</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">fF</font> = <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">g</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,<font color="Maroon" title="c1">D</font> <b>by </b><i><a class="ref" href="partfun1.html#T56" title="PARTFUN1:th.56">PARTFUN1:56</a></i>;<br/>
<a NAME="E4:86"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/>
<a NAME="E5:86"/><b>then </b><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">D</font> : <font color="Olive" title="b1">h</font> is <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> </span>}</span> 
 
;<br/>
<a NAME="E6:86"/><i><font color="Green" title="E53">A4</font></i>: 
<font color="Maroon" title="c6">g</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D2" title="UNIALG_1:def.2">UNIALG_1:def 2</a><br/>

<b>assume </b><a NAME="E1:86_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c7">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> )
 ;<br/>

<b>hence </b><a NAME="E2:86_1"/>
<font color="Maroon" title="c8">y</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>
 <b>by </b><i><a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:86"/><b>then </b><i><font color="Green" title="E54">A5</font></i>: 
<font color="Maroon" title="c6">g</font> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E5:86"><i><font color="Green" title="E52">A3</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:86_2"/>
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  ex <font color="Olive" title="b1">h</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">h</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="c4">F</font> &amp; <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( <font color="Maroon" title="c4">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">h</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">h</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="c4">F</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1"><b>suppose </b></a><a NAME="E1:86_2_1"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:86_2_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_2"><b>suppose </b></a><a NAME="E1:86_2_2"/>
<font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:86_2_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_6.html#T60" title="FUNCT_6:th.60">FUNCT_6:60</a></i>;<br/><b>hence </b><a NAME="E3:86_2_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E5:86"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="comput_1.html#E6:86"><i><font color="Green" title="E53">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_3"><b>suppose </b></a><a NAME="E1:86_2_3"/>
 ex <font color="Olive" title="b1">h</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">h</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="c4">F</font> &amp; <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:86_2_3"/><b>then </b>
<span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <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="comput_1.html#T10" target="_self" title="COMPUT_1:th.10">Th10</a></i>;<br/><b>hence </b><a NAME="E3:86_2_3"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E7:86"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_4"><b>suppose </b></a><a NAME="E1:86_2_4"/><i><font color="Green" title="E55">A6</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">h</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">h</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="c4">F</font> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 ;<br/><div class="add"><a NAME="E2:86_2_4"/><b>then </b>
for <font color="Olive" title="b1">h</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">h</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="c4">F</font> holds <br/>( <font color="Olive" title="b1">h</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> &amp; not <font color="Olive" title="b1">h</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#E2:86"><i><font color="Green" title="E51">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:86_2_4"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">F</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> is  <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>  <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:86"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="comput_1.html#E1:86_2_4"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="comput_1.html#T49" target="_self" title="COMPUT_1:th.49">Th49</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
