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

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

<b>set </b><font color="Maroon" title="c2">X</font> = <font color="Maroon" title="c1">D</font>;<br/>
<b>let </b><font color="Maroon" title="c3">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font>;<br/>

<b>assume </b><a NAME="E1:77"/><i><font color="Green" title="E44">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <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>  holds  <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:77"/><i><font color="Green" title="E45">A2</font></i>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">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="c3">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="E3:77"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">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> &amp;  <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>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E4:77"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c3">F</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">Y</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <font color="Maroon" title="c1">D</font></span>)</span> <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="E6:77"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c4">Y</font> is    <a href="rfunct_3.html#M1" title="RFUNCT_3:mode.1">PartFunc-set</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">f</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_3.html#D3" title="RFUNCT_3:def.3">RFUNCT_3:def 3</a><br/>

<a NAME="E1:77_1"/>
<font color="Maroon" title="c5">f</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="E2:77_1"/>
<font color="Maroon" title="c5">f</font> 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>
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E47">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/><a NAME="E1:77_2"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><div><i><font color="Green" title="E49">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</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> ;<br/><b>assume </b><a NAME="E1:77_2_1"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font>]
 ;<br/><a NAME="E2:77_2_1"/>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:77"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:77_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="comput_1.html#E1:77_2_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><a NAME="E3:77_2"/><i><font color="Green" title="E50">A8</font></i>: 
for <font color="Olive" title="b1">i</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="comput_1.html#E1:77_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="comput_1.html#E2:77_2"><i><font color="Green" title="E49">A6</font></i></a>);<br/></i><b>let </b><font color="Maroon" title="c6">m</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> ;<br/><b>assume </b><a NAME="E4:77_2"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:77_2"/><i><font color="Green" title="E51">A9</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E7:77_2"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E8:77_2"/>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:77_2"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="comput_1.html#E6:77_2"><i><font color="Green" title="E51">A9</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:77"/><i><font color="Green" title="E48">A10</font></i>: 
<font color="Maroon" title="c4">Y</font> is <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D1" target="_self" title="COMPUT_1:def.1">COMPUT_1:def 1</a><br/>

<b>assume </b><a NAME="E1:77_3"/>
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:77_3"/><i><font color="Green" title="E49">A11</font></i>: 
( <font color="Maroon" title="c7">i</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="c3">F</font> &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</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>assume </b><a NAME="E4:77_3"/>
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:77_3"/><i><font color="Green" title="E50">A12</font></i>: 
( <font color="Maroon" title="c8">j</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="c3">F</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</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="c9">i</font> = <font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c8">j</font> as    <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>  <b>by </b><i><a class="txt" href="comput_1.html#E3:77_3"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="comput_1.html#E6:77_3"><i><font color="Green" title="E50">A12</font></i></a></i>;<br/>
<a NAME="E8:77_3"/>
( <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> or <font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> )
 
;<br/>
<a NAME="E9:77_3"/><b>then </b>
( <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">g</font> or <font color="Maroon" title="c6">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:77"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="comput_1.html#E3:77_3"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="comput_1.html#E6:77_3"><i><font color="Green" title="E50">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:77_3"/>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="ref" href="partfun1.html#T135" title="PARTFUN1:th.135">PARTFUN1:135</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:77"/><b>then </b>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">F</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ,<font color="Maroon" title="c1">D</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E2:77"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="comput_1.html#E6:77"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="comput_1.html#T18" target="_self" title="COMPUT_1:th.18">Th18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">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="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="E11:77"/><i><font color="Green" title="E49">A13</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">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"> <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#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">Y</font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E2:77"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="comput_1.html#E8:77"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></i>;<br/>
<a NAME="E12:77"/>
<font color="Maroon" title="c5">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="77_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">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="E1:77_4"/>
<font color="Maroon" title="c6">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="c5">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">A1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:77_4"/><i><font color="Green" title="E50">A14</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A1</font> &amp; <font color="Maroon" title="c8">A1</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#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">Y</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="comput_1.html#E11:77"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">g1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">Y</font><b> such that </b><br/><a NAME="E5:77_4"/><i><font color="Green" title="E51">A15</font></i>: 
<font color="Maroon" title="c8">A1</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="c9">g1</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:77_4"><i><font color="Green" title="E50">A14</font></i></a></i>;<br/>
<b>assume </b><a NAME="E6:77_4"/>
<font color="Maroon" title="c7">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="c5">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">A2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:77_4"/><i><font color="Green" title="E52">A16</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c10">A2</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#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">Y</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="comput_1.html#E11:77"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">g2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">Y</font><b> such that </b><br/><a NAME="E10:77_4"/><i><font color="Green" title="E53">A17</font></i>: 
<font color="Maroon" title="c10">A2</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="c11">g2</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E8:77_4"><i><font color="Green" title="E52">A16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">i1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:77_4"/><i><font color="Green" title="E54">A18</font></i>: 
( <font color="Maroon" title="c12">i1</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="c3">F</font> &amp; <font color="Maroon" title="c9">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i1</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>consider </b><font color="Maroon" title="c13">i2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:77_4"/><i><font color="Green" title="E55">A19</font></i>: 
( <font color="Maroon" title="c13">i2</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="c3">F</font> &amp; <font color="Maroon" title="c11">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i2</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">i1</font> = <font color="Maroon" title="c12">i1</font>, <font color="Maroon" title="c15">i2</font> = <font color="Maroon" title="c13">i2</font> as    <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>  <b>by </b><i><a class="txt" href="comput_1.html#E12:77_4"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="comput_1.html#E14:77_4"><i><font color="Green" title="E55">A19</font></i></a></i>;<br/>
<a NAME="E16:77_4"/>
( <font color="Maroon" title="c14">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i2</font> or <font color="Maroon" title="c15">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i1</font> )
 
;<br/>
<a NAME="E17:77_4"/><b>then </b>
( <font color="Maroon" title="c9">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">g2</font> or <font color="Maroon" title="c11">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">g1</font> )
 
<b>by </b><i><a class="txt" href="comput_1.html#E7:77"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="comput_1.html#E12:77_4"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="comput_1.html#E14:77_4"><i><font color="Green" title="E55">A19</font></i></a></i>;<br/>
<a NAME="E18:77_4"/><b>then </b>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">g1</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> <font color="Maroon" title="c11">g2</font> or  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">g2</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> <font color="Maroon" title="c9">g1</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>
<b>hence </b><a NAME="E19:77_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">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="c7">y</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E3:77_4"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="comput_1.html#E5:77_4"><i><font color="Green" title="E51">A15</font></i></a>, <a class="txt" href="comput_1.html#E8:77_4"><i><font color="Green" title="E52">A16</font></i></a>, <a class="txt" href="comput_1.html#E10:77_4"><i><font color="Green" title="E53">A17</font></i></a>, <a class="ref" href="unialg_1.html#D1" title="UNIALG_1:def.1">UNIALG_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:77"/>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c3">F</font> <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/>


</div>
