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<b>let </b><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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green" title="E41">A1</font></i>: 
<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> 
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>
 ;<br/>

<b>set </b><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/>
<b>set </b><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="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" title="c1">R</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp; not <font color="Maroon" title="c2">S</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> ) or <font color="Maroon" title="c1">R</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> or <font color="Maroon" title="c2">S</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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" title="c1">R</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp; not <font color="Maroon" title="c2">S</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><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#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><b>consider </b><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><b> such that </b><br/><a NAME="E4:57_1_1"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:57"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/><a NAME="E5:57_1_1"/><i><font color="Green" title="E45">A5</font></i>: 
( <font color="Maroon" title="c7">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">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> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">Re</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> iff <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> ) ) )
 <b>by </b><i><a class="txt" href="dickson.html#E4:57_1_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">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" title="c5">Re</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a>
 <b>by </b><i><a class="txt" href="dickson.html#E3:57"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E2:57_1_1_1"/>
<font color="Maroon" title="c6">Se</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:57"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="waybel20.html#T15" title="WAYBEL20:th.15">WAYBEL20:15</a></i>;<br/><a NAME="E3:57_1_1_1"/>
<font color="Maroon" title="c5">Re</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</a>
 <b>by </b><i><a class="txt" href="dickson.html#E3:57"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:57_1_1_1"/>
<font color="Maroon" title="c6">Se</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:57"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="waybel20.html#T16" title="WAYBEL20:th.16">WAYBEL20:16</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:57_1_1"/><i><font color="Green" title="E46">A6</font></i>: 
<font color="Maroon" title="c2">S</font> 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="hs2(this)" href="javascript:()" title="57_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">t</font> be   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c6">Se</font>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="dickson.html#E5:57_1_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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">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="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1</span>)</span>;<br/><b>consider </b><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><b> such that </b><br/><a NAME="E3:57_1_1_2"/><i><font color="Green" title="E47">A7</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c10">sr</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</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="c11">sr</font> = <font color="Maroon" title="c10">sr</font> as   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c5">Re</font> ;<br/><b>consider </b><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:57_1_1_2"/><i><font color="Green" title="E48">A8</font></i>: 
<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>
 <b>and </b><br/><a NAME="E7:57_1_1_2"/><i><font color="Green" title="E49">A9</font></i>: 
<font color="Maroon" title="c11">sr</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</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="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:57"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="dickson.html#E3:57"><i><font color="Green" title="E43">A3</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="c14">i</font> = <font color="Maroon" title="c12">i</font>;<br/><b>take </b><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c13">j</font>;<br/><b>thus </b><a NAME="E8:57_1_1_2"/>
<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>
 <b>by </b><i><a class="txt" href="dickson.html#E6:57_1_1_2"><i><font color="Green" title="E48">A8</font></i></a></i>;<br/><i><font color="Green" title="E50">A10</font></i>: <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="normsp_1.html#K2" title="NORMSP_1:func.2">.</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="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="dickson.html#E3:57_1_1_2"><i><font color="Green" title="E47">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">t</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="dickson.html#E5:57_1_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>
;<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="normsp_1.html#K2" title="NORMSP_1:func.2">.</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="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="dickson.html#E3:57_1_1_2"><i><font color="Green" title="E47">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">t</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">j</font>
<b>by </b><i><a class="txt" href="dickson.html#E5:57_1_1"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>
;<br/><b>hence </b><a NAME="E11:57_1_1_2"/>
<font color="Maroon" title="c8">t</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</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="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:57_1_1"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="dickson.html#E7:57_1_1_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="txt" href="dickson.html#E9:57_1_1_2"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/></div><b>end;</b></div><a NAME="E9:57_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 <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> )
 <b>by </b><i><a class="txt" href="dickson.html#E7:57_1_1"><i><font color="Green" title="E46">A6</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="E10:57_1_1"/>
<font color="Maroon" title="c2">S</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 <b>by </b><i><a class="txt" href="dickson.html#E7:57_1_1"><i><font color="Green" title="E46">A6</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="57_1_2"><b>suppose </b></a><a NAME="E1:57_1_2"/><i><font color="Green" title="E44">A11</font></i>: 
( <font color="Maroon" title="c1">R</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> or <font color="Maroon" title="c2">S</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 ;<br/><div class="add"><div><i><font color="Green" title="E45">A12</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" title="c1">R</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> or <font color="Maroon" title="c2">S</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E1:57_1_2"><i><font color="Green" title="E44">A11</font></i></a></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" title="E46">A13</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c5">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><b> such that </b><br/><a NAME="E3:57_1_2_1_1_1"/><i><font color="Green" title="E47">A14</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:57"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/><b>thus </b><a NAME="E4:57_1_2_1_1_1"/>
<font color="Maroon" title="c2">S</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:57_1_2_1_1_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="dickson.html#E3:57_1_2_1_1_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">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" title="c2">S</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:57_1_2_1_1_2"/>
<font color="Maroon" title="c2">S</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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" 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>
 <b>by </b><i><a class="txt" href="dickson.html#E2:57_1_2"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><a NAME="E2:57_1_2_2"/><b>then </b>
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>
 <b>by </b><i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:57_1_2_2"/>
<font color="Maroon" title="c2">S</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a>
 <b>by </b><i><a class="ref" href="orders_2.html#D4" title="ORDERS_2:def.4">ORDERS_2:def 4</a></i>;<br/><a NAME="E4:57_1_2_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>
 <b>by </b><i><a class="txt" href="dickson.html#E2:57_1_2"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><a NAME="E5:57_1_2_2"/><b>then </b>
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>
 <b>by </b><i><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/><b>hence </b><a NAME="E6:57_1_2_2"/>
<font color="Maroon" title="c2">S</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</a>
 <b>by </b><i><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:57_1_2"/>
<font color="Maroon" title="c2">S</font> 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/><b>thus </b><a NAME="E5:57_1_2"/>
<font color="Maroon" title="c2">S</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:57_1_2"><i><font color="Green" title="E45">A12</font></i></a>, <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>
