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

<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">R</font>,<font color="Maroon">S</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">R</font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">R</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 ;<br/>

<b>set </b><font color="Maroon">CRS</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">IRS</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1"/>
( ( not <font color="Maroon">R</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">S</font> is <a href="struct_0.html#V3">empty</a> ) or <font color="Maroon">R</font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">S</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="57_1_1"><b>suppose </b></a><a NAME="E1:57_1_1"/>
( not <font color="Maroon">R</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">S</font> is <a href="struct_0.html#V3">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">Re</font> = <font color="Maroon">R</font>, <font color="Maroon">Se</font> = <font color="Maroon">S</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Re</font>,<font color="Maroon">Se</font><b> such that </b><br/><a NAME="E4:57_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/><a NAME="E5:57_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Se</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Re</font> holds <br/> ( <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">y</font> iff <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> ) ) )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:57_1_1_1"/>
<font color="Maroon">Re</font> is <a href="orders_2.html#V2">reflexive</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E2:57_1_1_1"/>
<font color="Maroon">Se</font> is <a href="orders_2.html#V2">reflexive</a>
 <b>by </b><i>, <a class="ref" href="waybel20.html#T15">WAYBEL20:15</a></i>;<br/><a NAME="E3:57_1_1_1"/>
<font color="Maroon">Re</font> is <a href="orders_2.html#V3">transitive</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:57_1_1_1"/>
<font color="Maroon">Se</font> is <a href="orders_2.html#V3">transitive</a>
 <b>by </b><i>, <a class="ref" href="waybel20.html#T16">WAYBEL20:16</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:57_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Se</font>;<br/><b>reconsider </b><font color="Maroon">fi</font> = <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Se</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Re</font> <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<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">Re</font> = <font color="Maroon">fi</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">sr</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">Re</font><b> such that </b><br/><a NAME="E3:57_1_1_2"/><i><font color="Green">E34</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">sr</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">sr</font> = <font color="Maroon">sr</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">Re</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="E6:57_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E7:57_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">sr</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">sr</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="txt" href="dickson.html#E4:57_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E8:57_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><i><font color="Green">E69</font></i>: <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">sr</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">t</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
;<br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">sr</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">t</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
;<br/><b>hence </b><a NAME="E11:57_1_1_2"/>
<font color="Maroon">t</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="dickson.html#D2" target="_self">Def2</a>, <a class="ref" href="dickson.html#T4" target="_self">Th4</a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/></div><b>end;</b></div><a NAME="E9:57_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 <font color="Maroon">Se</font> 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>, <a class="txt" href="dickson.html#E7:57_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E10:57_1_1"/>
<font color="Maroon">S</font> 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="57_1_2"><b>suppose </b></a><a NAME="E1:57_1_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">R</font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">S</font> is <a href="struct_0.html#V3">empty</a> )
 ;<br/><div class="add"><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_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="E1:57_1_2_1_1"/>
( <font color="Maroon">R</font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">S</font> is <a href="struct_0.html#V3">empty</a> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_2_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">R</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">R</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:57_1_2_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/><b>thus </b><a NAME="E4:57_1_2_1_1_1"/>
<font color="Maroon">S</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_2_1_1_2"/>
<font color="Maroon">S</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:57_1_2_1_1_2"/>
<font color="Maroon">S</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_2"><b>now </b></a><div class="add"><a NAME="E1:57_1_2_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E2:57_1_2_2"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_2"/>
<font color="Maroon">S</font> is <a href="orders_2.html#V2">reflexive</a>
 <b>by </b><i><a class="ref" href="orders_2.html#D4">ORDERS_2:def 4</a></i>;<br/><a NAME="E4:57_1_2_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E5:57_1_2_2"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/><b>hence </b><a NAME="E6:57_1_2_2"/>
<font color="Maroon">S</font> is <a href="orders_2.html#V3">transitive</a>
 <b>by </b><i><a class="ref" href="orders_2.html#D5">ORDERS_2:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:57_1_2"/>
<font color="Maroon">S</font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:57_1_2"/>
<font color="Maroon">S</font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
