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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">X</font> = <font color="Maroon">D</font>;<br/>
<b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font>;<br/>

<b>assume </b><a NAME="E1:79"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:79"/><i><font color="Green">E49</font></i>: 
 <a href="card_3.html#K3">Union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<a NAME="E3:79"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E4:79"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Y</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font></span>)</span> <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E6:79"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">Y</font> is    <a href="rfunct_3.html#M1">PartFunc-set</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:79_1"/>
<font color="Maroon">f</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font>
 
;<br/>
<b>hence </b><a NAME="E2:79_1"/>
<font color="Maroon">f</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">D</font>
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><a NAME="E1:79_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/><div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:79_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/><a NAME="E2:79_2_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:79_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="comput_1.html#E1:79_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><a NAME="E3:79_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="comput_1.html#E6:79"><i><font color="Green">E51</font></i></a>, <a class="txt" href="comput_1.html#E7:79"><i><font color="Green">E52</font></i></a>);<br/></i><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:79_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:79_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E7:79_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E8:79_2"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:79_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="comput_1.html#E2:79_2"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:79"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">Y</font> is <a href="comput_1.html#V1" target="_self">compatible</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:79_3"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:79_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>assume </b><a NAME="E4:79_3"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:79_3"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font>, <font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="comput_1.html#E7:79"><i><font color="Green">E52</font></i></a>, </i>;<br/>
<a NAME="E8:79_3"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> or <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 
;<br/>
<a NAME="E9:79_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> or <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="comput_1.html#T5" target="_self">Th5</a>, <a class="txt" href="comput_1.html#E7:79"><i><font color="Green">E52</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E10:79_3"/>
<font color="Maroon">f</font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="partfun1.html#T135">PARTFUN1:135</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:79"/><b>then </b>
 <a href="card_3.html#K3">Union</a> <font color="Maroon">F</font> is   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:79"><i><font color="Green">E48</font></i></a>, <a class="txt" href="comput_1.html#E2:79"><i><font color="Green">E49</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> =  <a href="card_3.html#K3">Union</a> <font color="Maroon">F</font> as    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">D</font> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<a NAME="E11:79"/><i><font color="Green">E67</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E1:79"><i><font color="Green">E48</font></i></a>, <a class="txt" href="comput_1.html#E6:79"><i><font color="Green">E51</font></i></a>, </i>;<br/>
<a NAME="E12:79"/>
<font color="Maroon">f</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">y</font> be   <a href="finseq_1.html#NM1">FinSequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="comput_1.html#D4" target="_self">COMPUT_1:def 4</a><br/>

<b>assume </b><a NAME="E1:79_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:79_4"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="comput_1.html#E7:79"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">g1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:79_4"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E6:79_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:79_4"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Olive">g</font></span>)</span> where <font color="Olive">g</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Y</font> : verum </span>}</span>  )
 <b>by </b><i><a class="txt" href="comput_1.html#E7:79"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">g2</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E10:79_4"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:79_4"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">i2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:79_4"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i1</font>, <font color="Maroon">i2</font> = <font color="Maroon">i2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E16:79_4"/>
( <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> or <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> )
 
;<br/>
<a NAME="E17:79_4"/><b>then </b>
( <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g2</font> or <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g1</font> )
 
<b>by </b><i><a class="ref" href="comput_1.html#T5" target="_self">Th5</a>, , </i>;<br/>
<a NAME="E18:79_4"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> or  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> )
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>
<b>hence </b><a NAME="E19:79_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="comput_1.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:79"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">D</font> : <font color="Olive">g</font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E14:79"/>
 <a href="card_3.html#K3">Union</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">D</font>
 ;<br/>


</div>
