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

<b>let </b><font color="Maroon">M</font> be    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">N</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">M</font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">N</font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 <b>and </b><br/><a NAME="E3:56"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">M</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 <b>and </b><br/><a NAME="E4:56"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">N</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">M'</font> = <font color="Maroon">M</font> as  <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">N'</font> = <font color="Maroon">N</font> as  <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E7:56"/>
( <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">M'</font>,<font color="Maroon">N'</font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="orders_2.html#V2">reflexive</a> &amp; <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">M'</font>,<font color="Maroon">N'</font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="orders_2.html#V3">transitive</a> )
 
;<br/>
<b>hence </b><a NAME="E8:56"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">N</font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1"/>
( ( not <font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">N</font> is <a href="struct_0.html#V3">empty</a> ) or <font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">N</font> is <a href="struct_0.html#V3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1"><b>suppose </b></a><a NAME="E1:56_1_1"/>
( not <font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">N</font> is <a href="struct_0.html#V3">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">Me</font> = <font color="Maroon">M</font>, <font color="Maroon">Ne</font> = <font color="Maroon">N</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/><b>set </b><font color="Maroon">CPMN</font> = <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">Me</font>,<font color="Maroon">Ne</font></span><a href="yellow_3.html#K3">:]</a></span>;<br/><a NAME="E3:56_1_1"/>
for <font color="Olive">f</font> being  <a href="normsp_1.html#NM2">sequence</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">Me</font>,<font color="Maroon">Ne</font></span><a href="yellow_3.html#K3">:]</a></span>  ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">f</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">f</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">j</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">Me</font>,<font color="Maroon">Ne</font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Me</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="yellow_3.html#K8">`1</a> ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Me</font><b> such that </b><br/><a NAME="E2:56_1_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">a</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Me</font> ;<br/>
<b>consider </b><font color="Maroon">sa</font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Me</font><b> such that </b><br/><a NAME="E5:56_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">sa</font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E6:56_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">sa</font> is <a href="dickson.html#V2" target="_self">weakly-ascending</a>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">NS</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E8:56_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">sa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">NS</font>
 <b>by </b><i>, <a class="ref" href="bhsp_3.html#D4">BHSP_3:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Ne</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="yellow_3.html#K9">`2</a> ;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Ne</font><b> such that </b><br/><a NAME="E10:56_1_1_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">b</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Ne</font> ;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:56_1_1_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">b</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, , <a class="txt" href="dickson.html#E4:56"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">i</font>
;<br/>

<b>take </b>
<font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">j</font>
;<br/>

<b>thus </b><a NAME="E14:56_1_1_1"/>
<font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="dickson.html#T4" target="_self">Th4</a>, <a class="ref" href="seqm_3.html#T7">SEQM_3:7</a></i>;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">i</font></span>)</span>, <font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">j</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">Me</font>,<font color="Maroon">Ne</font></span><a href="yellow_3.html#K3">:]</a></span> ;<br/>
<a NAME="E16:56_1_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">sa</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E72</font></i>: <font color="Maroon">sa</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">a</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a></i>
;<br/>
<i><font color="Green">E73</font></i>: <font color="Maroon">sa</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">a</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a></i>
;<br/>
<a NAME="E19:56_1_1_1"/>
<font color="Maroon">M</font> is <a href="orders_2.html#V3">transitive</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:56_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K8">`1</a> 
 
<b>by </b><i>, <a class="ref" href="dickson.html#T4" target="_self">Th4</a>, , <a class="ref" href="dickson.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E21:56_1_1_1"/>
( <font color="Maroon">b</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="yellow_3.html#K9">`2</a>  &amp; <font color="Maroon">b</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="yellow_3.html#K9">`2</a>  )
 
<b>by </b><i><a class="ref" href="dickson.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E22:56_1_1_1"/>
<font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="dickson.html#T4" target="_self">Th4</a>, , <a class="ref" href="yellow_3.html#T12">YELLOW_3:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:56_1_1"/><b>then </b>
for <font color="Olive">N</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">Me</font>,<font color="Maroon">Ne</font></span><a href="yellow_3.html#K3">:]</a></span> holds <br/> (  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Olive">N</font> is <a href="finset_1.html#V1">finite</a> &amp; not  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Olive">N</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:56_1_1"/>
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">N</font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="dickson.html#V4" target="_self">Dickson</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2"><b>suppose </b></a><a NAME="E1:56_1_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">N</font> is <a href="struct_0.html#V3">empty</a> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_2_1_1"/>
( <font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">N</font> is <a href="struct_0.html#V3">empty</a> )
 </span><b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_2_1_1_1"/>
<font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">M2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> as  <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:56_1_2_1_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">M2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><b>hence </b><a NAME="E4:56_1_2_1_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_2_1_1_2"/>
<font color="Maroon">N</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">N2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> as  <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:56_1_2_1_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,<font color="Maroon">N2</font></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><b>hence </b><a NAME="E4:56_1_2_1_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:56_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">N</font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><a NAME="E4:56_1_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">N</font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>hence </b><a NAME="E5:56_1_2"/>
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">N</font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="dickson.html#V4" target="_self">Dickson</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
