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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">D</font>, <font color="Maroon" title="c3">E</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="c4">X</font> = <font color="Maroon" title="c2">D</font>;<br/>
<b>set </b><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c3">E</font>;<br/>
<b>let </b><font color="Maroon" title="c6">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">D</font>, <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c3">E</font>;<br/>

<b>assume </b><a NAME="E1:76"/><i><font color="Green" title="E45">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> is <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a>
 ;<br/>

<b>assume </b><a NAME="E2:76"/><i><font color="Green" title="E46">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> holds   <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c3">E</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1"/>
( <font color="Maroon" title="c6">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c6">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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1"><b>suppose </b></a><a NAME="E1:76_1_1"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c6">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c7">f</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  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="c3">E</font> <b>by </b><i><a class="ref" href="comput_1.html#T30" target="_self" title="COMPUT_1:th.30">Th30</a></i>;<br/><b>take </b>
<font color="Maroon" title="c7">f</font>
;<br/><b>thus </b><a NAME="E3:76_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:76_1_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/><b>thus </b><a NAME="E4:76_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c3">E</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>thus </b><a NAME="E5:76_1_1"/>
verum
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_2"><b>suppose </b></a><a NAME="E1:76_1_2"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c6">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> 
 ;<br/><div class="add"><a NAME="E2:76_1_2"/><i><font color="Green" title="E48">A5</font></i>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>;<br/><a NAME="E3:76_1_2"/><i><font color="Green" title="E49">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> is <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_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="fraenkel.html#D1" title="FRAENKEL:def.1">FRAENKEL:def 1</a><br/>

<b>assume </b><a NAME="E1:76_1_2_1"/><i><font color="Green" title="E50">A7</font></i>: 
<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> <font color="Maroon" title="c6">F</font>
 ;<br/>

<a NAME="E2:76_1_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">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="c3">E</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E3:76_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">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="c3">E</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:76_1_2_1"><i><font color="Green" title="E50">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:76_1_2_1"/>
<font color="Maroon" title="c7">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c7">f</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">F</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="comput_1.html#E1:76"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="comput_1.html#E2:76_1_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="comput_1.html#T14" target="_self" title="COMPUT_1:th.14">Th14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">rngF</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a> <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="comput_1.html#E1:76"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="comput_1.html#E1:76_1_2"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="comput_1.html#E3:76_1_2"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><b>set </b><font color="Maroon" title="c9">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">rngF</font> : verum </span>}</span> ;<br/><a NAME="E6:76_1_2"/><i><font color="Green" title="E50">A8</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c3">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_2"><b>proof </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> ; <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="E1:76_1_2_2"/>
<font color="Maroon" title="c10">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> <font color="Maroon" title="c7">f</font>
 ;<br/>

<a NAME="E2:76_1_2_2"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">rngF</font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E2:76_1_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:76_1_2_2"/><i><font color="Green" title="E51">A9</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">d</font> &amp; <font color="Maroon" title="c11">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">rngF</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">rngF</font><b> such that </b><br/><a NAME="E6:76_1_2_2"/><i><font color="Green" title="E52">A10</font></i>: 
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">g</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:76_1_2_2"><i><font color="Green" title="E51">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:76_1_2_2"/><i><font color="Green" title="E53">A11</font></i>: 
( <font color="Maroon" title="c13">e</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">F</font> &amp; <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</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/>
<b>reconsider </b><font color="Maroon" title="c14">e</font> = <font color="Maroon" title="c13">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> <b>by </b><i><a class="txt" href="comput_1.html#E8:76_1_2_2"><i><font color="Green" title="E53">A11</font></i></a></i>;<br/>
<a NAME="E10:76_1_2_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">e</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c3">E</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E2:76"><i><font color="Green" title="E46">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:76_1_2_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c3">E</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:76_1_2_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="txt" href="comput_1.html#E6:76_1_2_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="comput_1.html#E8:76_1_2_2"><i><font color="Green" title="E53">A11</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:76_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c3">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">E</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="E8:76_1_2"/><b>then </b><i><font color="Green" title="E51">A12</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">E</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E6:76_1_2"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:76_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_3"><b>proof </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> ; <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="E1:76_1_2_3"/>
<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> <font color="Maroon" title="c7">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:76_1_2_3"/><i><font color="Green" title="E52">A13</font></i>: 
( <font color="Maroon" title="c11">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> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</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="E4:76_1_2_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">rngF</font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E2:76_1_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="comput_1.html#E3:76_1_2_3"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></i>;<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="E6:76_1_2_3"/><i><font color="Green" title="E53">A14</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d</font> &amp; <font color="Maroon" title="c12">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">g</font></span>)</span> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">rngF</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">rngF</font><b> such that </b><br/><a NAME="E8:76_1_2_3"/><i><font color="Green" title="E54">A15</font></i>: 
<font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E6:76_1_2_3"><i><font color="Green" title="E53">A14</font></i></a></i>;<br/>
<a NAME="E9:76_1_2_3"/>
( <font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">rngF</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">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="c3">E</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>then reconsider </b><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">g</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="c3">E</font> ;<br/>
<a NAME="E11:76_1_2_3"/><i><font color="Green" title="E55">A16</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E2:76_1_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="comput_1.html#E6:76_1_2_3"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="comput_1.html#E8:76_1_2_3"><i><font color="Green" title="E54">A15</font></i></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th16</a></i>;<br/>
<a NAME="E12:76_1_2_3"/><i><font color="Green" title="E56">A17</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">E</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E13:76_1_2_3"/>
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="c14">g</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E6:76_1_2_3"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="comput_1.html#E8:76_1_2_3"><i><font color="Green" title="E54">A15</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E14:76_1_2_3"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">E</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:76_1_2_3"><i><font color="Green" title="E52">A13</font></i></a>, <a class="txt" href="comput_1.html#E11:76_1_2_3"><i><font color="Green" title="E55">A16</font></i></a>, <a class="txt" href="comput_1.html#E12:76_1_2_3"><i><font color="Green" title="E56">A17</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c7">f</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c3">E</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,<font color="Maroon" title="c3">E</font> <b>by </b><i><a class="txt" href="comput_1.html#E8:76_1_2"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">f</font> as    <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="c3">E</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c3">E</font> <b>by </b><i><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/><a NAME="E12:76_1_2"/>
<font color="Maroon" title="c11">f</font> is <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font>, <font color="Maroon" title="c13">y</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D1" title="UNIALG_1:def.1">UNIALG_1:def 1</a><br/>

<b>assume </b><a NAME="E1:76_1_2_4"/>
( <font color="Maroon" title="c12">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="c11">f</font> &amp; <font color="Maroon" title="c13">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="c11">f</font> )
 ;<br/>

<a NAME="E2:76_1_2_4"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E6:76_1_2"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<b>hence </b><a NAME="E3:76_1_2_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">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="c13">y</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E13:76_1_2"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</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="c3">E</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c3">E</font> : <font color="Olive" title="b1">g</font> is <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a> </span>}</span> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c12">f</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">F</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="c3">E</font> ;<br/><b>take </b>
<font color="Maroon" title="c12">f</font>
;<br/><b>thus </b><a NAME="E15:76_1_2"/>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">F</font>
 ;<br/><b>thus </b><a NAME="E16:76_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c3">E</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E6:76_1_2"><i><font color="Green" title="E50">A8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
