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

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V16">connected</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">x</font> be    <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">y</font> be    <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>;<br/>



<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">FOAR</font> =  <a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font>;<br/>
<a NAME="E1:60"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <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">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> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:60"/><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:60_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:60_1"/><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:60_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</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">n</font> = <font color="Maroon">n</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="60_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</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="60_1_1_1"><b>suppose </b></a><a NAME="E1:60_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then 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="E3:60_1_1_1"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <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="E4:60_1_1_1"/><i><font color="Green">E49</font></i>: 
( <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>, , </i>;<br/><a NAME="E5:60_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E6:60_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">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="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">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> ) or ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">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="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">y</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="Maroon">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">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="Maroon">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>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2"><b>suppose </b></a><a NAME="E1:60_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:60_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1 is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><a NAME="E3:60_1_1_2"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</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">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> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, </i>;<br/><b>then 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="E5:60_1_1_2"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <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="E6:60_1_1_2"/><i><font color="Green">E64</font></i>: 
( <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> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:60_1_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E8:60_1_1_2"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:60_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:60_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">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="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">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> ) or ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">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="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">y</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="Maroon">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">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="Maroon">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>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:60"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">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="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">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> ) or ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">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="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">y</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="Maroon">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">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="Maroon">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>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span></span>)</span> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">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="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">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> ) or ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">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="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">y</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="Maroon">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">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="Maroon">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>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span></span>)</span> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_1"><b>suppose </b></a><a NAME="E1:60_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:60_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:60_2_1"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:60_2_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_2"><b>suppose </b></a><a NAME="E1:60_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">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="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">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> )
 ;<br/><div class="add"><a NAME="E2:60_2_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:60_2_2"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:60_2_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_2_3"><b>suppose </b></a><a NAME="E1:60_2_3"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">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="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">y</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="Maroon">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">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="Maroon">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>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span></span>)</span> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">NEXTXY</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="Maroon">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">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="Maroon">y</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span>;<br/><b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:60_2_3"/><i><font color="Green">E84</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="Maroon">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">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="Maroon">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">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:60_2_3"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E7:60_2_3"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <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> 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">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> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E8:60_2_3"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <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> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="bagorder.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E9:60_2_3"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span> <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> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:60_2_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="bagorder.html#K12" target="_self">FinOrd-Approx</a> <font color="Maroon">R</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
