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

<b>set </b><font color="Maroon">IR</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">CR</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">FBCP</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<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> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">y</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> ) </span>}</span> ;<br/>
<a NAME="E1:59_1_1"/><i><font color="Green">E30</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>]
 
;<br/>
<a NAME="E2:59_1_1"/><i><font color="Green">E31</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>, <font color="Olive">y1</font>, <font color="Olive">y2</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">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:59_1_1"/><i><font color="Green">E36</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="E5:59_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">y</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E6:59_1_1"/><i><font color="Green">E48</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="recdef_1.html#S1">RECDEF_1:sch 1</a>(, <a class="txt" href="bagorder.html#E4:59_1_1"><i><font color="Green">E36</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:59_1_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>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:59_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_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="E1:59_1_1_1_1"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">F0</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> ) ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:59_1_1_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> ) ) </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:59_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E4:59_1_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> ) )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:59_1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:59_1_1_1_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E4:59_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:59_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="bagorder.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_1_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:59_1_1_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x'</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><b>set </b><font color="Maroon">FX</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> ) </span>}</span> ;<br/><a NAME="E4:59_1_1_1_1_2"/><i><font color="Green">E69</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:59_1_1_1_1_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> ) </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:59_1_1_1_1_2_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:59_1_1_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E6:59_1_1_1_1_2"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : ( <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E7:59_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E9:59_1_1"/>
(  <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>  &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">y</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> ) ) </span>}</span>  &amp; ( 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">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">y</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> ) </span>}</span>  ) )
 <b>by </b><i>, <a class="txt" href="bagorder.html#E5:59_1_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>


</div>
