<?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  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="E1:85"/><i><font color="Green" title="E48">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> &amp; not <font color="Maroon" title="c4">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>and </b><br/><a NAME="E2:85"/><i><font color="Green" title="E49">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> &amp; not <font color="Olive" title="b1">h</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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>set </b><font color="Maroon" title="c6">n</font> =  <a href="comput_1.html#K3" title="COMPUT_1:func.3">arity</a> <font color="Maroon" title="c4">F</font>;<br/>
<a NAME="E3:85"/><i><font color="Green" title="E50">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">arity</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T46" target="_self" title="COMPUT_1:th.46">Th46</a></i>;<br/>
<a NAME="E4:85"/><i><font color="Green" title="E51">A4</font></i>: 
<span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">arity</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><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></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:85_1"/><i><font color="Green" title="E52">A5</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">arity</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>
 ;<br/>

<a NAME="E2:85_1"/><i><font color="Green" title="E53">A6</font></i>: 
<span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">arity</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:85_1_1"/><i><font color="Green" title="E54">A7</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">arity</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>
 ;<br/>

<i><font color="Green" title="E55">A8</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</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="funct_6.html#K4" title="FUNCT_6:func.4">meet</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#T49" title="FUNCT_6:th.49">FUNCT_6:49</a></i>
<br/>.= 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#D4" title="FUNCT_6:def.4">FUNCT_6:def 4</a></i>
;<br/>
<a NAME="E3:85_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:85"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:85_1_1"/><i><font color="Green" title="E56">A9</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E6:85_1_1"/>
( <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</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;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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#E5:85_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:85_1_1"/><b>then </b><i><font color="Green" title="E57">A10</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> is    <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/>
<a NAME="E8:85_1_1"/><i><font color="Green" title="E58">A11</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#D2" title="FUNCT_6:def.2">FUNCT_6:def 2</a></i>;<br/>
<a NAME="E9:85_1_1"/><b>then </b>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="comput_1.html#E5:85_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="txt" href="comput_1.html#E7:85_1_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="funct_6.html#T28" title="FUNCT_6:th.28">FUNCT_6:28</a></i>;<br/>
<a NAME="E10:85_1_1"/><b>then </b><i><font color="Green" title="E59">A12</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:85_1_1_2"/>
<font color="Maroon" title="c10">y</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> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:85_1_1_2"/><i><font color="Green" title="E60">A13</font></i>: 
( <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:85_1_1_2"/><i><font color="Green" title="E61">A14</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E8:85_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="comput_1.html#E3:85_1_1_2"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="funct_6.html#T28" title="FUNCT_6:th.28">FUNCT_6:28</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">Fw</font> = <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">w</font> as    <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="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/><i><font color="Green" title="E62">A15</font></i>: <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">w</font> = 
 <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">w</font></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E8:85_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="comput_1.html#E3:85_1_1_2"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="funct_6.html#D2" title="FUNCT_6:def.2">FUNCT_6:def 2</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">Fw</font>
<b>by </b><i><a class="ref" href="funct_5.html#T21" title="FUNCT_5:th.21">FUNCT_5:21</a></i>
;<br/><a NAME="E7:85_1_1_2"/><i><font color="Green" title="E63">A16</font></i>: 
<font color="Maroon" title="c12">Fw</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>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:85_1_1_2"><i><font color="Green" title="E61">A14</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E8:85_1_1_2"/><b>then </b>
( not <font color="Maroon" title="c12">Fw</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c12">Fw</font> is <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a> )
 <b>by </b><i><a class="txt" href="comput_1.html#E2:85"><i><font color="Green" title="E49">A2</font></i></a></i>;<br/><a NAME="E9:85_1_1_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">Fw</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c12">Fw</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="ref" href="comput_1.html#T25" target="_self" title="COMPUT_1:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E10:85_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:85_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="comput_1.html#E3:85_1_1_2"><i><font color="Green" title="E60">A13</font></i></a>, <a class="txt" href="comput_1.html#E6:85_1_1_2"><i><font color="Green" title="E62">A15</font></i></a>, <a class="txt" href="comput_1.html#E7:85_1_1_2"><i><font color="Green" title="E63">A16</font></i></a>, <a class="ref" href="comput_1.html#D7" target="_self" title="COMPUT_1:def.7">Def7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:85_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</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>
 <b>by </b><i><a class="txt" href="comput_1.html#E2:85_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="comput_1.html#E10:85_1_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:85_1"/><i><font color="Green" title="E54">A17</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T25" target="_self" title="COMPUT_1:th.25">Th25</a></i>;<br/>
<a NAME="E4:85_1"/><i><font color="Green" title="E55">A18</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T49" title="FUNCT_6:th.49">FUNCT_6:49</a></i>;<br/>
<a NAME="E5:85_1"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:85_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:85_1_2"/><i><font color="Green" title="E56">A19</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c4">F</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:85_1_2"/><i><font color="Green" title="E57">A20</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c4">F</font></span>)</span> holds <br/><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E5:85_1_2"/><i><font color="Green" title="E58">A21</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#D3" title="FUNCT_6:def.3">FUNCT_6:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:85_1_2_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:85_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="funct_6.html#T28" title="FUNCT_6:th.28">FUNCT_6:28</a></i>;<br/><b>hence </b><a NAME="E3:85_1_2_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:85_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span>
 ;<br/><a NAME="E3:85_1_2_1"/><b>then </b><i><font color="Green" title="E59">A22</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:85_1_2_1"/><b>then </b>
( <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</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;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E5:85_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> is    <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>hence </b><a NAME="E6:85_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:85_1_2_1"><i><font color="Green" title="E59">A22</font></i></a>, <a class="ref" href="funct_6.html#T28" title="FUNCT_6:th.28">FUNCT_6:28</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:85_1_2"/><b>then </b><i><font color="Green" title="E59">A23</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">g</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="comput_1.html#E3:85_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="comput_1.html#E5:85_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E9:85_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:85_1_2_2"/>
<font color="Maroon" title="c11">x</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="c10">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:85_1_2_2"/><i><font color="Green" title="E60">A24</font></i>: 
( <font color="Maroon" title="c12">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">g</font> &amp; <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:85_1_2_2"/><i><font color="Green" title="E61">A25</font></i>: 
<font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:85_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="comput_1.html#E4:85_1_2"><i><font color="Green" title="E57">A20</font></i></a>, <a class="txt" href="comput_1.html#E3:85_1_2_2"><i><font color="Green" title="E60">A24</font></i></a></i>;<br/>
<a NAME="E5:85_1_2_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">Fd</font> = <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font> as    <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#E3:85_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="comput_1.html#E5:85_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="comput_1.html#E7:85_1_2"><i><font color="Green" title="E59">A23</font></i></a>, <a class="txt" href="comput_1.html#E3:85_1_2_2"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="finseq_2.html#T13" title="FINSEQ_2:th.13">FINSEQ_2:13</a></i>;<br/>
<i><font color="Green" title="E62">A26</font></i>: <span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font> = 
 <a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">d</font></span>)</span>
<b>by </b><i><a class="txt" href="comput_1.html#E3:85_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="comput_1.html#E5:85_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="comput_1.html#E3:85_1_2_2"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="funct_6.html#D3" title="FUNCT_6:def.3">FUNCT_6:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">Fd</font>
<b>by </b><i><a class="ref" href="funct_5.html#T21" title="FUNCT_5:th.21">FUNCT_5:21</a></i>
;<br/>
<a NAME="E8:85_1_2_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">Fd</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E9:85_1_2_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:85_1_2_2"><i><font color="Green" title="E60">A24</font></i></a>, <a class="txt" href="comput_1.html#E4:85_1_2_2"><i><font color="Green" title="E61">A25</font></i></a>, <a class="txt" href="comput_1.html#E7:85_1_2_2"><i><font color="Green" title="E62">A26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c10">g</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E11:85_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">g</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>by </b><i><a class="txt" href="comput_1.html#E3:85_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="txt" href="comput_1.html#E5:85_1_2"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="comput_1.html#E7:85_1_2"><i><font color="Green" title="E59">A23</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:85_1_2"/><b>then </b>
<font color="Maroon" title="c8">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:85_1_2"><i><font color="Green" title="E56">A19</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>hence </b><a NAME="E13:85_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:85_1"/><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</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> <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> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:85_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="comput_1.html#E2:85_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="comput_1.html#E4:85_1"><i><font color="Green" title="E55">A18</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E7:85_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><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></span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:85"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="comput_1.html#E1:85_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="comput_1.html#E2:85_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="comput_1.html#E3:85_1"><i><font color="Green" title="E54">A17</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:85"/><b>then </b><i><font color="Green" title="E52">A27</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">arity</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:85"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E6:85"/>
<span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">arity</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="ref" href="msualg_1.html#T12" title="MSUALG_1:th.12">MSUALG_1:12</a></i>;<br/>
<a NAME="E7:85"/><b>then </b><i><font color="Green" title="E53">A28</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:85"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:85"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="ref" href="comput_1.html#T46" target="_self" title="COMPUT_1:th.46">Th46</a></i>;<br/>
<a NAME="E9:85"/><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> is    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</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="txt" href="comput_1.html#E7:85"><i><font color="Green" title="E53">A28</font></i></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E10:85"/><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> 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>
 
<b>by </b><i><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/>
<a NAME="E11:85"/><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#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">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> as    <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/>
<a NAME="E13:85"/>
<font color="Maroon" title="c7">fF</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="85_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">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="E1:85_2"/><i><font color="Green" title="E54">A29</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c9">y</font> &amp; <font color="Maroon" title="c8">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="c7">fF</font> )
 ;<br/>

<a NAME="E2:85_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K3" title="COMPUT_1:func.3">arity</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E3:85"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="comput_1.html#E1:85_2"><i><font color="Green" title="E54">A29</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:85_2"/><b>then </b>
<font color="Maroon" title="c9">y</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">arity</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:85_2"><i><font color="Green" title="E54">A29</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>hence </b><a NAME="E4:85_2"/>
<font color="Maroon" title="c9">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="c7">fF</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E5:85"><i><font color="Green" title="E52">A27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:85"/>
<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  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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#E3:85"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="comput_1.html#E4:85"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>

<b>thus </b><a NAME="E15:85"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K3" title="COMPUT_1:func.3">arity</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:85"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="comput_1.html#E4:85"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
