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

<span class="kw">let </span><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  &amp; <font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>  &amp; <font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>  implies ( <font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>  ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <span class="kw">and </span><br/><a NAME="E2:58"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a> 
 <span class="kw">and </span><br/><a NAME="E3:58"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">CRS</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">IRS</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_1"/>
( ( not <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; not <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  ) or <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <font color="Maroon" title="c2">S</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="58_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1"/>
( not <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; not <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>  )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">Re</font> = <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c6">Se</font> = <font color="Maroon" title="c2">S</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/><span class="kw">consider </span><font color="Maroon" title="c7">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">Re</font>,<font color="Maroon" title="c6">Se</font><span class="kw"> such that </span><br/><a NAME="E4:58_1_1"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c7">f</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></span>;<br/><a NAME="E5:58_1_1"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></span>;<br/><a NAME="E6:58_1_1"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">Se</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></span>;<br/><a NAME="E7:58_1_1"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c5">Re</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></span>;<br/><a NAME="E8:58_1_1"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<font color="Maroon" title="c5">Re</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></span>;<br/><a NAME="E9:58_1_1"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
<font color="Maroon" title="c6">Se</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E7:58_1_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="waybel20.html#T15" title="WAYBEL20:th.15">WAYBEL20:15</a></span>;<br/><a NAME="E10:58_1_1"/>
<font color="Maroon" title="c6">Se</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E8:58_1_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="waybel20.html#T16" title="WAYBEL20:th.16">WAYBEL20:16</a></span>;<br/><span class="kw">hence </span><a NAME="E11:58_1_1"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
<font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58_1_1"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">t</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c6">Se</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">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="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">j</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">fi</font> = <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">Se</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Re</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E6:58_1_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></span>;<br/><span class="kw">deffunc </span><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> ) <span class="kw">-&gt; </span>   <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">Re</font> = <font color="Maroon" title="c9">fi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">sr</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">Re</font><span class="kw"> such that </span><br/><a NAME="E3:58_1_1_1"/><span class="lab"><font color="Green" title="E51">A11</font></span>: 
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="c10">sr</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c11">sr</font> = <font color="Maroon" title="c10">sr</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c5">Re</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">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> <span class="kw"> such that </span><br/><a NAME="E6:58_1_1_1"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">and </span><br/><a NAME="E7:58_1_1_1"/><span class="lab"><font color="Green" title="E53">A13</font></span>: 
<font color="Maroon" title="c11">sr</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">sr</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="dickson.html#T30" target="_self" title="DICKSON:th.30">Th30</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c12">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">j</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c13">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font> )</span><br/><span class="kw">thus </span><a NAME="E8:58_1_1_1"/>
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_1_1"><span class="lab"><font color="Green" title="E52">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font></span><br/><span class="lab"><font color="Green" title="E54">A14</font></span>: <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">sr</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font></span>)</span> = 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E6:58_1_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></span>
;<br/><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">sr</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font></span>)</span> = 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1"><span class="lab"><font color="Green" title="E51">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E6:58_1_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></span>
;<br/><span class="kw">hence </span><a NAME="E11:58_1_1_1"/>
<font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E7:58_1_1_1"><span class="lab"><font color="Green" title="E53">A13</font></span></a>, <a class="txt" href="#E9:58_1_1_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E13:58_1_1"/><span class="kw">then </span>
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 <font color="Maroon" title="c6">Se</font> 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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="dickson.html#T32" target="_self" title="DICKSON:th.32">Th32</a></span>;<br/><span class="kw">hence </span><a NAME="E14:58_1_1"/>
<font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58_1_1"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="dickson.html#T33" target="_self" title="DICKSON:th.33">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_2"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
( <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>  )</span><br/><div class="add"><div><span class="lab"><font color="Green" title="E45">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_1_2_1_1"/>
( <font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_2_1_1_1"/><span class="lab"><font color="Green" title="E46">A17</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> </span><br/><div class="add"><a NAME="E2:58_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font> st <font color="Olive" title="b1">f</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_1_2_1_1_1"/>
<font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_2_1_1_1"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_2_1_1_2"/>
<font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_1_2_1_1_2"/>
<font color="Maroon" title="c2">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E3:58_1_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font>
 ;<br/><a NAME="E4:58_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A18</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></span>;<br/><a NAME="E5:58_1_2"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a></span>;<br/><a NAME="E6:58_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A19</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></span>;<br/><a NAME="E7:58_1_2"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
<font color="Maroon" title="c2">S</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_2"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="ref" href="orders_2.html#D4" title="ORDERS_2:def.4">ORDERS_2:def 4</a></span>;<br/><a NAME="E8:58_1_2"/>
<font color="Maroon" title="c2">S</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_1_2"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E9:58_1_2"/>
<font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_1_2"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a> </span><br/><span class="kw">thus </span><a NAME="E10:58_1_2"/>
<font color="Maroon" title="c2">S</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="dickson.html#T37" target="_self" title="DICKSON:th.37">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
