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

<b>let </b><font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c2">N</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c1">M</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c2">N</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 <b>and </b><br/><a NAME="E3:56"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c1">M</font> is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>
 <b>and </b><br/><a NAME="E4:56"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c2">N</font> is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">M'</font> = <font color="Maroon" title="c1">M</font> as  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="dickson.html#E3:56"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">N'</font> = <font color="Maroon" title="c2">N</font> as  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="dickson.html#E4:56"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></i>;<br/>
<a NAME="E7:56"/>
( <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c3">M'</font>,<font color="Maroon" title="c4">N'</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> &amp; <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c3">M'</font>,<font color="Maroon" title="c4">N'</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> )
 
;<br/>
<b>hence </b><a NAME="E8:56"/><i><font color="Green" title="E44">A5</font></i>: 
<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>
 <b>by </b><i><a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></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" title="c1">M</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Maroon" title="c2">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> ) or <font color="Maroon" title="c1">M</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> or <font color="Maroon" title="c2">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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" title="c1">M</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not <font color="Maroon" title="c2">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">Me</font> = <font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c6">Ne</font> = <font color="Maroon" title="c2">N</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><b>set </b><font color="Maroon" title="c7">CPMN</font> = <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c5">Me</font>,<font color="Maroon" title="c6">Ne</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/><a NAME="E3:56_1_1"/>
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c5">Me</font>,<font color="Maroon" title="c6">Ne</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">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" title="c8">f</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c5">Me</font>,<font color="Maroon" title="c6">Ne</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Me</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> ;<br/>
<b>consider </b><font color="Maroon" title="c9">a</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Me</font><b> such that </b><br/><a NAME="E2:56_1_1_1"/><i><font color="Green" title="E45">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c9">a</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c5">Me</font> ;<br/>
<b>consider </b><font color="Maroon" title="c11">sa</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c5">Me</font><b> such that </b><br/><a NAME="E5:56_1_1_1"/><i><font color="Green" title="E46">A7</font></i>: 
<font color="Maroon" title="c11">sa</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c10">a</font>
 <b>and </b><br/><a NAME="E6:56_1_1_1"/><i><font color="Green" title="E47">A8</font></i>: 
<font color="Maroon" title="c11">sa</font> is <a href="dickson.html#V2" title="DICKSON:attr.2">weakly-ascending</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:56"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="dickson.html#T36" target="_self" title="DICKSON:th.36">Th36</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">NS</font> being  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E8:56_1_1_1"/><i><font color="Green" title="E48">A9</font></i>: 
<font color="Maroon" title="c11">sa</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c10">a</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">NS</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:56_1_1_1"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="bhsp_3.html#D4" title="BHSP_3:def.4">BHSP_3:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">Ne</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1</span>)</span></span>)</span> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a> ;<br/>
<b>consider </b><font color="Maroon" title="c13">b</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">Ne</font><b> such that </b><br/><a NAME="E10:56_1_1_1"/><i><font color="Green" title="E49">A10</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c13">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c14">b</font> = <font color="Maroon" title="c13">b</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c6">Ne</font> ;<br/>
<b>consider </b><font color="Maroon" title="c15">i</font>, <font color="Maroon" title="c16">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E13:56_1_1_1"/><i><font color="Green" title="E50">A11</font></i>: 
( <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font> &amp; <font color="Maroon" title="c14">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">j</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E2:56"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="dickson.html#E4:56"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="dickson.html#T30" target="_self" title="DICKSON:th.30">Th30</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">j</font>
;<br/>

<b>thus </b><a NAME="E14:56_1_1_1"/>
<font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="dickson.html#E13:56_1_1_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="seqm_3.html#T7" title="SEQM_3:th.7">SEQM_3:7</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span>, <font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c5">Me</font>,<font color="Maroon" title="c6">Ne</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> ;<br/>
<a NAME="E16:56_1_1_1"/><i><font color="Green" title="E51">A12</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">sa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E52">A13</font></i>: <font color="Maroon" title="c11">sa</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font> = 
<font color="Maroon" title="c10">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span>
<b>by </b><i><a class="txt" href="dickson.html#E8:56_1_1_1"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span></span>)</span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
<b>by </b><i><a class="txt" href="dickson.html#E2:56_1_1_1"><i><font color="Green" title="E45">A6</font></i></a></i>
;<br/>
<i><font color="Green" title="E53">A14</font></i>: <font color="Maroon" title="c11">sa</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">j</font> = 
<font color="Maroon" title="c10">a</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span>
<b>by </b><i><a class="txt" href="dickson.html#E8:56_1_1_1"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="dickson.html#E16:56_1_1_1"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
<b>by </b><i><a class="txt" href="dickson.html#E2:56_1_1_1"><i><font color="Green" title="E45">A6</font></i></a></i>
;<br/>
<a NAME="E19:56_1_1_1"/>
<font color="Maroon" title="c1">M</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 
<b>by </b><i><a class="txt" href="dickson.html#E3:56"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></i>;<br/>
<a NAME="E20:56_1_1_1"/><b>then </b><i><font color="Green" title="E54">A15</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c18">y</font> <a href="yellow_3.html#K8" title="YELLOW_3:func.8">`1</a> 
 
<b>by </b><i><a class="txt" href="dickson.html#E6:56_1_1_1"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="dickson.html#E13:56_1_1_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="dickson.html#E17:56_1_1_1"><i><font color="Green" title="E52">A13</font></i></a>, <a class="txt" href="dickson.html#E18:56_1_1_1"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="dickson.html#T4" target="_self" title="DICKSON:th.4">Th4</a></i>;<br/>
<a NAME="E21:56_1_1_1"/>
( <font color="Maroon" title="c14">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  &amp; <font color="Maroon" title="c14">b</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font> <a href="yellow_3.html#K9" title="YELLOW_3:func.9">`2</a>  )
 
<b>by </b><i><a class="txt" href="dickson.html#E10:56_1_1_1"><i><font color="Green" title="E49">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:56_1_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">NS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E13:56_1_1_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="dickson.html#E20:56_1_1_1"><i><font color="Green" title="E54">A15</font></i></a>, <a class="ref" href="yellow_3.html#T12" title="YELLOW_3:th.12">YELLOW_3:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:56_1_1"/><b>then </b>
for <font color="Olive" title="b1">N</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c5">Me</font>,<font color="Maroon" title="c6">Ne</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> holds <br/> (  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Olive" title="b1">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; not  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Olive" title="b1">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E8:56"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="dickson.html#T32" target="_self" title="DICKSON:th.32">Th32</a></i>;<br/><b>hence </b><a NAME="E5:56_1_1"/>
<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 <b>by </b><i><a class="txt" href="dickson.html#E8:56"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="dickson.html#T33" target="_self" title="DICKSON:th.33">Th33</a></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" title="E45">A16</font></i>: 
( <font color="Maroon" title="c1">M</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> or <font color="Maroon" title="c2">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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" title="c1">M</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> or <font color="Maroon" title="c2">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E1:56_1_2"><i><font color="Green" title="E45">A16</font></i></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" title="c1">M</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">M2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">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" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">M2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><b>hence </b><a NAME="E4:56_1_2_1_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="c2">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">N2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">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" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c5">N2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><b>hence </b><a NAME="E4:56_1_2_1_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">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" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">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" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">N</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 <b>by </b><i><a class="ref" href="dickson.html#T37" target="_self" title="DICKSON:th.37">Th37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
