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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D</font>, <font color="Olive" title="b2">E</font> being 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/> for <font color="Olive" title="b3">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">D</font>,<span class="p1">(<span class="default"><a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Olive" title="b2">E</font></span>)</span>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">F</font> is  <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a>  &amp; ( for <font color="Olive" title="b4">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">D</font> holds   <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">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="Olive" title="b2">E</font> ) holds <br/> ex <font color="Olive" title="b4">f</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Olive" title="b2">E</font> st <br/>( <font color="Olive" title="b4">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="Olive" title="b3">F</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">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="Olive" title="b2">E</font> )</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">D</font>,<span class="p1">(<span class="default"><a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Maroon" title="c3">E</font></span>)</span>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">F</font> is  <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a>  &amp; ( for <font color="Olive" title="b2">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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">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> ) holds <br/> ex <font color="Olive" title="b2">f</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Maroon" title="c3">E</font> st <br/>( <font color="Olive" title="b2">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="Olive" title="b1">F</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">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> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c2">D</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c3">E</font>;<br/>
<span class="kw">let </span><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>,<span class="p1">(<span class="default"><a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Maroon" title="c3">E</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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>  &amp; ( 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#NK2" title="RELAT_1:NK.2">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> ) implies  ex <font color="Olive" title="b1">f</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Maroon" title="c3">E</font> st <br/>( <font color="Olive" title="b1">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> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">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> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:78"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
 <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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <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> st not  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> or  ex <font color="Olive" title="b1">f</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Maroon" title="c3">E</font> st <br/>( <font color="Olive" title="b1">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> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">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> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:78"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
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#NK2" title="RELAT_1:NK.2">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Maroon" title="c3">E</font> st <br/>( <font color="Olive" title="b1">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> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">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> )</span><br/>

<a NAME="E3:78"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> is  <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funct_1.html#D19" title="FUNCT_1:def.19">FUNCT_1:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or <font color="Maroon" title="c7">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<a NAME="E1:78_1"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
 <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#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Maroon" title="c3">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:78_1"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<span class="kw">hence </span><a NAME="E3:78_1"/>
<font color="Maroon" title="c7">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">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="funct_1.html#V4" title="FUNCT_1:attr.4">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E42">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c7">rngF</font> : verum </span>}</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E44">A4</font></span></a></span>;<br/>
<a NAME="E6:78"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">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="78_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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="c9">f</font> or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:78_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> <font color="Maroon" title="c9">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:78_2"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">and </span><br/><a NAME="E4:78_2"/><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<font color="Maroon" title="c9">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:78_2"/>
<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#NK2" title="RELAT_1:NK.2">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="c7">rngF</font> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_2"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:78_2"/><span class="lab"><font color="Green" title="E48">A10</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:78_2"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
<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#NK2" title="RELAT_1:NK.2">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="c7">rngF</font> : verum </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><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="c7">rngF</font><span class="kw"> such that </span><br/><a NAME="E10:78_2"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
<font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">g</font>
 <span class="kw">and </span><br/><a NAME="E11:78_2"/>
verum
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78_2"><span class="lab"><font color="Green" title="E49">A11</font></span></a></span>;<br/>
<a NAME="E12:78_2"/><span class="lab"><font color="Green" title="E51">A13</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rngF</font>
 
;<br/>
<a NAME="E13:78_2"/>
 <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#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Maroon" title="c3">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Maroon" title="c3">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:78_2"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>;<br/>
<a NAME="E15:78_2"/><span class="lab"><font color="Green" title="E52">A14</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:78_2"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E10:78_2"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E16:78_2"/><span class="lab"><font color="Green" title="E53">A15</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E17:78_2"/>
<font color="Maroon" title="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:78_2"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E10:78_2"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:78_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_2"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="txt" href="#E16:78_2"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E15:78_2"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:78"/><span class="lab"><font color="Green" title="E46">A16</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">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="78_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">f</font> or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:78_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:78_3"/><span class="kw">then </span>
<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#NK2" title="RELAT_1:NK.2">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="c7">rngF</font> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:78_3"/><span class="lab"><font color="Green" title="E47">A17</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:78_3"/><span class="lab"><font color="Green" title="E48">A18</font></span>: 
<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#NK2" title="RELAT_1:NK.2">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="c7">rngF</font> : verum </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><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="c7">rngF</font><span class="kw"> such that </span><br/><a NAME="E7:78_3"/><span class="lab"><font color="Green" title="E49">A19</font></span>: 
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">g</font>
 <span class="kw">and </span><br/><a NAME="E8:78_3"/>
verum
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:78_3"><span class="lab"><font color="Green" title="E48">A18</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:78_3"/><span class="lab"><font color="Green" title="E50">A20</font></span>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">and </span><br/><a NAME="E11:78_3"/><span class="lab"><font color="Green" title="E51">A21</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:78_3"><span class="lab"><font color="Green" title="E50">A20</font></span></a></span>;<br/>
<a NAME="E13:78_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E43">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:78_3"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_3"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="txt" href="#E7:78_3"><span class="lab"><font color="Green" title="E49">A19</font></span></a>, <a class="txt" href="#E11:78_3"><span class="lab"><font color="Green" title="E51">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:78"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T162" title="FINSEQ_2:th.162">FINSEQ_2:162</a></span>;<br/>
<a NAME="E9:78"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:78"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></span>;<br/>
<a NAME="E12:78"/>
<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="78_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:78_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">f</font> is  <a href="card_3.html#V2" title="CARD_3:attr.2">with_common_domain</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:78"><span class="lab"><font color="Green" title="E46">A16</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="unialg_1.html#D1" title="UNIALG_1:def.1">UNIALG_1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:78"/><span class="kw">then </span>
<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/>
<span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="COMPUT_1:func.2">HFuncs</a> <font color="Maroon" title="c3">E</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">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> )</span><br/>

<span class="kw">thus </span><a NAME="E15:78"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K1" title="RELSET_1:func.1">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></span><br/>

<span class="kw">thus </span><a NAME="E16:78"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:78"><span class="lab"><font color="Green" title="E46">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
