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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>(  ex <font color="Olive">A'</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">d'</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">A'</font>,<font color="Maroon">L</font> ex <font color="Olive">Aq</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">dq</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">Aq</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Aq</font>,<font color="Olive">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Olive">A'</font>,<font color="Olive">d'</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">A'</font>,<font color="Olive">d'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">Aq</font>,<font color="Olive">dq</font></span><a href="tarski.html#K4">]</a></span> ) or ( <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">A'</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">d'</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">A'</font>,<font color="Maroon">L</font><br/> for <font color="Olive">Aq</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">dq</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">Aq</font>,<font color="Maroon">L</font>  holds <br/>(  not <font color="Olive">Aq</font>,<font color="Olive">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Olive">A'</font>,<font color="Olive">d'</font> or  not <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">A'</font>,<font color="Olive">d'</font></span><a href="tarski.html#K4">]</a></span> ) ) ) );<br/>
<a NAME="E1:73_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1"><b>proof </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>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_1_1_1"/>
(  ex <font color="Olive">A'</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">d'</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">A'</font>,<font color="Maroon">L</font> ex <font color="Olive">Aq</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">dq</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">Aq</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Aq</font>,<font color="Olive">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Olive">A'</font>,<font color="Olive">d'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">A'</font>,<font color="Olive">d'</font></span><a href="tarski.html#K4">]</a></span> ) or for <font color="Olive">A'</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">d'</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">A'</font>,<font color="Maroon">L</font><br/> for <font color="Olive">Aq</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">dq</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">Aq</font>,<font color="Maroon">L</font>  holds <br/>(  not <font color="Olive">Aq</font>,<font color="Olive">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Olive">A'</font>,<font color="Olive">d'</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">A'</font>,<font color="Olive">d'</font></span><a href="tarski.html#K4">]</a></span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:73_1_1_1_1_1"/>
 ex <font color="Olive">A'</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">d'</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">A'</font>,<font color="Maroon">L</font> ex <font color="Olive">Aq</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">dq</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">Aq</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Aq</font>,<font color="Olive">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Olive">A'</font>,<font color="Olive">d'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">A'</font>,<font color="Olive">d'</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">A'</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d'</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A'</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:73_1_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">Aq</font>,<font color="Maroon">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A'</font>,<font color="Maroon">d'</font>
 <b>and </b><br/><a NAME="E4:73_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A'</font>,<font color="Maroon">d'</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><b>take </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Aq</font>,<font color="Maroon">dq</font></span><a href="tarski.html#K4">]</a></span>
;<br/><b>thus </b><a NAME="E5:73_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Aq</font>,<font color="Maroon">dq</font></span><a href="tarski.html#K4">]</a></span>]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:73_1_1_1_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">A'</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">d'</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">A'</font>,<font color="Maroon">L</font><br/> for <font color="Olive">Aq</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">dq</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">Aq</font>,<font color="Maroon">L</font>  holds <br/>(  not <font color="Olive">Aq</font>,<font color="Olive">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Olive">A'</font>,<font color="Olive">d'</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">A'</font>,<font color="Olive">d'</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><div class="add"><b>take </b>
0
;<br/><b>thus </b><a NAME="E2:73_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,0]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:73_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E4:73_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:73_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">n</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">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="lattice5.html#S1" target="_self">LATTICE5:sch 1</a>(<a class="txt" href="lattice5.html#E10:73_1_1"><i><font color="Green">E53</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E6:73_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E7:73_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">A'</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">d'</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">A'</font>,<font color="Maroon">L</font> ex <font color="Olive">Aq</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">dq</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">Aq</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Aq</font>,<font color="Olive">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Olive">A'</font>,<font color="Olive">d'</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">A'</font>,<font color="Olive">d'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">Aq</font>,<font color="Olive">dq</font></span><a href="tarski.html#K4">]</a></span> );<br/>
<a NAME="E8:73_1_1"/>
 ex <font color="Olive">A'</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">d'</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">A'</font>,<font color="Maroon">L</font> ex <font color="Olive">Aq</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">dq</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">Aq</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">Aq</font>,<font color="Olive">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Olive">A'</font>,<font color="Olive">d'</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">A'</font>,<font color="Olive">d'</font></span><a href="tarski.html#K4">]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>consider </b><font color="Maroon">q</font> being    <a href="lattice5.html#M1" target="_self">QuadrSeq</a> of <font color="Maroon">d</font>;<br/>
<b>set </b><font color="Maroon">Aq</font> =  <a href="lattice5.html#K14" target="_self">NextSet</a> <font color="Maroon">d</font>;<br/>
<b>consider </b><font color="Maroon">dq</font> being   <a href="lattice5.html#NM4">distance_function</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K14" target="_self">NextSet</a> <font color="Maroon">d</font></span>)</span>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:73_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">dq</font> <a href="funct_2.html#R2">=</a>  <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">q</font>
 ;<br/>
<b>take </b>
 <a href="lattice5.html#K14" target="_self">NextSet</a> <font color="Maroon">d</font>
;<br/>

<b>take </b>
<font color="Maroon">dq</font>
;<br/>

<b>thus </b><a NAME="E4:73_1_1_2"/>
 <a href="lattice5.html#K14" target="_self">NextSet</a> <font color="Maroon">d</font>,<font color="Maroon">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A</font>,<font color="Maroon">d</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E3:73_1_1_2"><i><font color="Green">E51</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:73_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:73_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E10:73_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">A'</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d'</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A'</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq</font>,<font color="Maroon">L</font><b> such that </b><a NAME="E1:73_1_1_3"/>
<font color="Maroon">Aq</font>,<font color="Maroon">dq</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A'</font>,<font color="Maroon">d'</font>
 <b>and </b><br/><a NAME="E2:73_1_1_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A'</font>,<font color="Maroon">d'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E3:73_1_1_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Aq</font>,<font color="Maroon">dq</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E4:73_1_1_3"/>
 ex <font color="Olive">A1</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">d1</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">A1</font>,<font color="Maroon">L</font> ex <font color="Olive">AQ</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">DQ</font> being  <a href="lattice5.html#NM4">distance_function</a> of <font color="Olive">AQ</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">AQ</font>,<font color="Olive">DQ</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Olive">A1</font>,<font color="Olive">d1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">A1</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_3_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">Aq</font>
;<br/>

<b>take </b>
<font color="Maroon">dq</font>
;<br/>

<b>set </b><font color="Maroon">AQ</font> =  <a href="lattice5.html#K14" target="_self">NextSet</a> <font color="Maroon">dq</font>;<br/>
<b>consider </b><font color="Maroon">Q</font> being    <a href="lattice5.html#M1" target="_self">QuadrSeq</a> of <font color="Maroon">dq</font>;<br/>
<b>set </b><font color="Maroon">DQ</font> =  <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">Q</font>;<br/>
<b>take </b>
 <a href="lattice5.html#K14" target="_self">NextSet</a> <font color="Maroon">dq</font>
;<br/>

<b>take </b>
 <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">Q</font>
;<br/>

<b>thus </b><a NAME="E2:73_1_1_3_1"/>
 <a href="lattice5.html#K14" target="_self">NextSet</a> <font color="Maroon">dq</font>, <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">Q</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">Aq</font>,<font color="Maroon">dq</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E3:73_1_1_3_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Aq</font>,<font color="Maroon">dq</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:73_1_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E11:73_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div>
