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

<b>let </b><font color="Maroon" title="c1">R</font> be  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>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green" title="E35">A2</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font>  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> )
 ;<br/>

<b>set </b><font color="Maroon" title="c2">IR</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c3">CR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">IR'</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>;<br/>
<a NAME="E3:49"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c1">R</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#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">N</font> be  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="c1">R</font>;<br/>

<b>assume </b><a NAME="E4:49"/><i><font color="Green" title="E37">A4</font></i>: 
(  not  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c5">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c5">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1"/>
(  not  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c5">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> or  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c5">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E4:49"><i><font color="Green" title="E37">A4</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>suppose </b></a><a NAME="E1:49_1_1"/><i><font color="Green" title="E38">A5</font></i>: 
not  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c5">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">MCN</font> =  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c5">N</font> as  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>consider </b><font color="Maroon" title="c7">f</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> , <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c5">N</font><b> such that </b><br/><a NAME="E4:49_1_1"/><i><font color="Green" title="E39">A6</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>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="dickson.html#T3" target="_self" title="DICKSON:th.3">Th3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/><div><i><font color="Green" title="E40">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:49_1_1_1"/><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c9">x'</font> = <font color="Maroon" title="c8">x</font> as    <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>by </b><i><a class="txt" href="dickson.html#E1:49_1_1_1"><i><font color="Green" title="E41">A8</font></i></a></i>;<br/><a NAME="E3:49_1_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c5">N</font>
 ;<br/><a NAME="E4:49_1_1_1"/><b>then </b><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">MCN</font>
 ;<br/><a NAME="E5:49_1_1_1"/><b>then </b>
not <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</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#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="dickson.html#T23" target="_self" title="DICKSON:th.23">Th23</a></i>;<br/><a NAME="E6:49_1_1_1"/><b>then </b>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x'</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 ;<br/><b>hence </b><a NAME="E7:49_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">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="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:49_1_1_1"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c8">g</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="c1">R</font><b> such that </b><br/><a NAME="E7:49_1_1"/><i><font color="Green" title="E41">A10</font></i>: 
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>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="dickson.html#E5:49_1_1"><i><font color="Green" title="E40">A7</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>consider </b><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">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="E10:49_1_1"/><i><font color="Green" title="E42">A11</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">j</font>
 <b>and </b><br/><a NAME="E11:49_1_1"/><i><font color="Green" title="E43">A12</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">j</font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:49"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">gi</font> = <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c13">gj</font> = <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> ;<br/><a NAME="E13:49_1_1"/><i><font color="Green" title="E44">A13</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c12">gi</font>,<font color="Maroon" title="c13">gj</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:49_1_1"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E14:49_1_1"/><i><font color="Green" title="E45">A14</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">MCN</font>
 ;<br/><a NAME="E15:49_1_1"/><b>then </b><i><font color="Green" title="E46">A15</font></i>: 
not <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</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#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="dickson.html#T23" target="_self" title="DICKSON:th.23">Th23</a></i>;<br/><a NAME="E16:49_1_1"/><i><font color="Green" title="E47">A16</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">MCN</font>
 ;<br/><a NAME="E17:49_1_1"/><b>then </b><i><font color="Green" title="E48">A17</font></i>: 
not <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">j</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#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="dickson.html#T23" target="_self" title="DICKSON:th.23">Th23</a></i>;<br/><a NAME="E18:49_1_1"/><i><font color="Green" title="E49">A18</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E7:49_1_1"><i><font color="Green" title="E41">A10</font></i></a></i>;<br/><a NAME="E19:49_1_1"/><i><font color="Green" title="E50">A19</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E7:49_1_1"><i><font color="Green" title="E41">A10</font></i></a></i>;<br/><a NAME="E20:49_1_1"/><i><font color="Green" title="E51">A20</font></i>: 
<font color="Maroon" title="c13">gj</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c5">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="dickson.html#E16:49_1_1"><i><font color="Green" title="E47">A16</font></i></a>, <a class="txt" href="dickson.html#E17:49_1_1"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="dickson.html#E18:49_1_1"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="dickson.html#T20" target="_self" title="DICKSON:th.20">Th20</a></i>;<br/><a NAME="E21:49_1_1"/>
<font color="Maroon" title="c12">gi</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c5">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="dickson.html#E14:49_1_1"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="dickson.html#E15:49_1_1"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="dickson.html#E19:49_1_1"><i><font color="Green" title="E50">A19</font></i></a>, <a class="ref" href="dickson.html#T20" target="_self" title="DICKSON:th.20">Th20</a></i>;<br/><a NAME="E22:49_1_1"/><b>then </b><i><font color="Green" title="E52">A21</font></i>: 
<font color="Maroon" title="c12">gi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D26" title="WAYBEL_4:def.26">WAYBEL_4:def 26</a></i>;<br/><div><i><font color="Green" title="E53">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_2_1"/>
( <font color="Maroon" title="c12">gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">gj</font> or <font color="Maroon" title="c12">gi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">gj</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_1"/>
<font color="Maroon" title="c12">gi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">gj</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_1_2_1_1"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c12">gi</font>,<font color="Maroon" title="c13">gj</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> 
 <b>by </b><i><a class="txt" href="dickson.html#E13:49_1_1"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_2"/><i><font color="Green" title="E54">A23</font></i>: 
<font color="Maroon" title="c12">gi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">gj</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_2_1_2_1"/>
not <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c12">gi</font>,<font color="Maroon" title="c13">gj</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> 
 ;<br/><a NAME="E2:49_1_1_2_1_2_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c12">gi</font>,<font color="Maroon" title="c13">gj</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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="c1">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E13:49_1_1"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:49_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E20:49_1_1"><i><font color="Green" title="E51">A20</font></i></a>, <a class="txt" href="dickson.html#E22:49_1_1"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_1_2_1_2"><i><font color="Green" title="E54">A23</font></i></a>, <a class="ref" href="waybel_4.html#D26" title="WAYBEL_4:def.26">WAYBEL_4:def 26</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1_1_2_1_2"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c12">gi</font>,<font color="Maroon" title="c13">gj</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E24:49_1_1"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c12">gi</font>,<font color="Maroon" title="c13">gj</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:49_1_1"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E25:49_1_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c12">gi</font>,<font color="Maroon" title="c13">gj</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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="c1">R</font> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E23:49_1_1"><i><font color="Green" title="E53">A22</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E26:49_1_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c12">gi</font>,<font color="Maroon" title="c13">gj</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="dickson.html#D4" target="_self" title="DICKSON:def.4">Def4</a></i>;<br/><a NAME="E27:49_1_1"/><b>then </b>
<font color="Maroon" title="c12">gi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c13">gj</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><a NAME="E28:49_1_1"/><b>then </b><i><font color="Green" title="E54">A24</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c13">gj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c12">gi</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">mj</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span><b> such that </b><br/><a NAME="E30:49_1_1"/>
<font color="Maroon" title="c14">mj</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c5">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>and </b><br/><a NAME="E31:49_1_1"/><i><font color="Green" title="E55">A25</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c14">mj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E16:49_1_1"><i><font color="Green" title="E47">A16</font></i></a>, <a class="ref" href="dickson.html#D8" target="_self" title="DICKSON:def.8">Def8</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">mi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span><b> such that </b><br/><a NAME="E33:49_1_1"/>
<font color="Maroon" title="c15">mi</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c5">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>and </b><br/><a NAME="E34:49_1_1"/><i><font color="Green" title="E56">A26</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c15">mi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E14:49_1_1"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="dickson.html#D8" target="_self" title="DICKSON:def.8">Def8</a></i>;<br/><a NAME="E35:49_1_1"/>
<font color="Maroon" title="c13">gj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c14">mj</font>
 <b>by </b><i><a class="txt" href="dickson.html#E17:49_1_1"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="dickson.html#E18:49_1_1"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="dickson.html#E31:49_1_1"><i><font color="Green" title="E55">A25</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E36:49_1_1"/><b>then </b><i><font color="Green" title="E57">A27</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c13">gj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c14">mj</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/><a NAME="E37:49_1_1"/>
<font color="Maroon" title="c12">gi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c15">mi</font>
 <b>by </b><i><a class="txt" href="dickson.html#E15:49_1_1"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="dickson.html#E19:49_1_1"><i><font color="Green" title="E50">A19</font></i></a>, <a class="txt" href="dickson.html#E34:49_1_1"><i><font color="Green" title="E56">A26</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E38:49_1_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">j</font>
 <b>by </b><i><a class="txt" href="dickson.html#E28:49_1_1"><i><font color="Green" title="E54">A24</font></i></a>, <a class="txt" href="dickson.html#E31:49_1_1"><i><font color="Green" title="E55">A25</font></i></a>, <a class="txt" href="dickson.html#E34:49_1_1"><i><font color="Green" title="E56">A26</font></i></a>, <a class="txt" href="dickson.html#E36:49_1_1"><i><font color="Green" title="E57">A27</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/><b>hence </b><a NAME="E39:49_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="dickson.html#E4:49_1_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="dickson.html#E10:49_1_1"><i><font color="Green" title="E42">A11</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><b>suppose </b></a><a NAME="E1:49_1_2"/><i><font color="Green" title="E38">A28</font></i>: 
 <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c5">N</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> $2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,$2</span>)</span> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> $2</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,$2</span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:49_1_2"/><i><font color="Green" title="E39">A29</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E4:49_1_2"/><i><font color="Green" title="E40">A30</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <font color="Maroon" title="c5">N</font>
 <b>and </b><br/><a NAME="E5:49_1_2"/><i><font color="Green" title="E41">A31</font></i>: 
for <font color="Olive" title="b1">n</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="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S3" title="RECDEF_1:sch.3">RECDEF_1:sch 3</a>();<br/></i><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>;<br/><a NAME="E6:49_1_2"/><i><font color="Green" title="E42">A32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="dickson.html#E4:49_1_2"><i><font color="Green" title="E40">A30</font></i></a></i>;<br/><div><i><font color="Green" title="E43">A33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <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> ;<br/><b>assume </b><a NAME="E1:49_1_2_1"/><i><font color="Green" title="E44">A34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">i</font>]
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">fi</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_2_1"><i><font color="Green" title="E44">A34</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c9">IC</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">fi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c8">fi</font></span>)</span>;<br/><a NAME="E3:49_1_2_1"/><i><font color="Green" title="E45">A35</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:49_1_2"><i><font color="Green" title="E41">A31</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">fi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c8">fi</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><a NAME="E2:49_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c8">fi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c8">fi</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:49_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c8">fi</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c5">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_2_1"><i><font color="Green" title="E44">A34</font></i></a>, <a class="ref" href="dickson.html#T31" target="_self" title="DICKSON:th.31">Th31</a></i>;<br/><b>hence </b><a NAME="E4:49_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_2"><i><font color="Green" title="E38">A28</font></i></a>, <a class="ref" href="dickson.html#T22" target="_self" title="DICKSON:th.22">Th22</a></i>;<br/></div><b>end;</b></div><a NAME="E5:49_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:49_1_2_1"><i><font color="Green" title="E45">A35</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E6:49_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E8:49_1_2"/><i><font color="Green" title="E44">A36</font></i>: 
for <font color="Olive" title="b1">i</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">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="dickson.html#E6:49_1_2"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="dickson.html#E7:49_1_2"><i><font color="Green" title="E43">A33</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:49_1_2_2"/><i><font color="Green" title="E45">A37</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><a NAME="E2:49_1_2_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="txt" href="dickson.html#E8:49_1_2"><i><font color="Green" title="E44">A36</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_2_2"><i><font color="Green" title="E45">A37</font></i></a></i>;<br/><b>hence </b><a NAME="E3:49_1_2_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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="c1">R</font>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="dickson.html#E3:49_1_2"><i><font color="Green" title="E39">A29</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/><div><i><font color="Green" title="E45">A38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">i</font> be    <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> ;<br/><b>defpred </b><font color="Maroon">S<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>means </b>( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 implies <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 );<br/><a NAME="E1:49_1_2_3"/><i><font color="Green" title="E46">A39</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E2:49_1_2_3"/><i><font color="Green" title="E47">A40</font></i>: 
for <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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</font> be    <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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:49_1_2_3_1"/><i><font color="Green" title="E48">A41</font></i>: 
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font> implies <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">j</font> )
 <b>and </b><br/><a NAME="E2:49_1_2_3_1"/><i><font color="Green" title="E49">A42</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<a NAME="E3:49_1_2_3_1"/><i><font color="Green" title="E50">A43</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E2:49_1_2_3_1"><i><font color="Green" title="E49">A42</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">fj</font> = <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">j</font>, <font color="Maroon" title="c11">fj1</font> = <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="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c12">IC</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font></span>)</span>;<br/>
<a NAME="E5:49_1_2_3_1"/><i><font color="Green" title="E51">A44</font></i>: 
<font color="Maroon" title="c10">fj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E8:49_1_2"><i><font color="Green" title="E44">A36</font></i></a></i>;<br/>
<a NAME="E6:49_1_2_3_1"/><i><font color="Green" title="E52">A45</font></i>: 
<font color="Maroon" title="c11">fj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E5:49_1_2"><i><font color="Green" title="E41">A31</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><a NAME="E2:49_1_2_3_1_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:49_1_2_3_1_1"/><b>then </b>
<font color="Maroon" title="c10">fj</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c5">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="dickson.html#E5:49_1_2_3_1"><i><font color="Green" title="E51">A44</font></i></a>, <a class="ref" href="dickson.html#T31" target="_self" title="DICKSON:th.31">Th31</a></i>;<br/><b>hence </b><a NAME="E4:49_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_2"><i><font color="Green" title="E38">A28</font></i></a>, <a class="ref" href="dickson.html#T22" target="_self" title="DICKSON:th.22">Th22</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:49_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c11">fj1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E6:49_1_2_3_1"><i><font color="Green" title="E52">A45</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E9:49_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c11">fj1</font> <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="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:49_1_2_3_1"/><b>then </b><i><font color="Green" title="E53">A46</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c11">fj1</font>,<font color="Maroon" title="c10">fj</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</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="c1">R</font>
 
<b>by </b><i><a class="ref" href="wellord1.html#D1" title="WELLORD1:def.1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E11:49_1_2_3_1"/><b>then </b><i><font color="Green" title="E54">A47</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <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="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_3_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_2_3_1_2"/>
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font> or <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E3:49_1_2_3_1"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_3_1_2_1"><b>suppose </b></a><a NAME="E1:49_1_2_3_1_2_1"/>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_2_3_1_2_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <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="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:49"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_2_3_1"><i><font color="Green" title="E48">A41</font></i></a>, <a class="txt" href="dickson.html#E11:49_1_2_3_1"><i><font color="Green" title="E54">A47</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_3_1_2_2"><b>suppose </b></a><a NAME="E1:49_1_2_3_1_2_2"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_2_3_1_2_2"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <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="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E10:49_1_2_3_1"><i><font color="Green" title="E53">A46</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E3:49_1_2_3"/>
for <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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="dickson.html#E1:49_1_2_3"><i><font color="Green" title="E46">A39</font></i></a>, <a class="txt" href="dickson.html#E2:49_1_2_3"><i><font color="Green" title="E47">A40</font></i></a>);</i><br/></div><b>end;</b></div><div><i><font color="Green" title="E46">A48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">i</font> be    <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> ;<br/><b>defpred </b><font color="Maroon">S<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>means </b>( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 implies  not <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</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> $1 );<br/><a NAME="E1:49_1_2_4"/><i><font color="Green" title="E47">A49</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E2:49_1_2_4"/><i><font color="Green" title="E48">A50</font></i>: 
for <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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</font> be    <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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:49_1_2_4_1"/>
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font> implies  not <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</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="Maroon" title="c9">j</font> )
 <b>and </b><br/><a NAME="E2:49_1_2_4_1"/><i><font color="Green" title="E49">A51</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<a NAME="E3:49_1_2_4_1"/><i><font color="Green" title="E50">A52</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E2:49_1_2_4_1"><i><font color="Green" title="E49">A51</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">fj</font> = <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">j</font>, <font color="Maroon" title="c11">fj1</font> = <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="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> ;<br/>
<a NAME="E5:49_1_2_4_1"/><i><font color="Green" title="E51">A53</font></i>: 
<font color="Maroon" title="c10">fj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E8:49_1_2"><i><font color="Green" title="E44">A36</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_2_4_1_1"/>
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font> or <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E3:49_1_2_4_1"><i><font color="Green" title="E50">A52</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_2_4_1_1_1"/><i><font color="Green" title="E52">A54</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:49_1_2_4_1_1_1"/><i><font color="Green" title="E53">A55</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E3:49_1_2_4_1_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:49_1_2_4_1_1_1"/><b>then </b><i><font color="Green" title="E54">A56</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">j</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <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="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E11:49_1_2"><i><font color="Green" title="E45">A38</font></i></a></i>;<br/><a NAME="E5:49_1_2_4_1_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:49_1_2"><i><font color="Green" title="E45">A38</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_2_4_1_1_1"><i><font color="Green" title="E52">A54</font></i></a></i>;<br/><a NAME="E6:49_1_2_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:49"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="dickson.html#E2:49_1_2_4_1_1_1"><i><font color="Green" title="E53">A55</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><a NAME="E7:49_1_2_4_1_1_1"/><b>then </b><i><font color="Green" title="E55">A57</font></i>: 
<font color="Maroon" title="c11">fj1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="dickson.html#E4:49_1_2_4_1_1_1"><i><font color="Green" title="E54">A56</font></i></a>, <a class="ref" href="dickson.html#T9" target="_self" title="DICKSON:th.9">Th9</a></i>;<br/><b>set </b><font color="Maroon" title="c12">IC</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font></span>)</span>;<br/><a NAME="E8:49_1_2_4_1_1_1"/><i><font color="Green" title="E56">A58</font></i>: 
<font color="Maroon" title="c11">fj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:49_1_2"><i><font color="Green" title="E41">A31</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_4_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_4_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><a NAME="E2:49_1_2_4_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:49_1_2_4_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">fj</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c5">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="dickson.html#E5:49_1_2_4_1"><i><font color="Green" title="E51">A53</font></i></a>, <a class="ref" href="dickson.html#T31" target="_self" title="DICKSON:th.31">Th31</a></i>;<br/><b>hence </b><a NAME="E4:49_1_2_4_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_2"><i><font color="Green" title="E38">A28</font></i></a>, <a class="ref" href="dickson.html#T22" target="_self" title="DICKSON:th.22">Th22</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:49_1_2_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E7:49_1_2_4_1_1_1"><i><font color="Green" title="E55">A57</font></i></a>, <a class="txt" href="dickson.html#E8:49_1_2_4_1_1_1"><i><font color="Green" title="E56">A58</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_2_4_1_1_2"/><i><font color="Green" title="E52">A59</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:49_1_2_4_1_1_2"/><i><font color="Green" title="E53">A60</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E3:49_1_2_4_1_1_2"/>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:49_1_2_4_1_1_2"/><b>then </b>
<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="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:49_1_2"><i><font color="Green" title="E45">A38</font></i></a></i>;<br/><a NAME="E5:49_1_2_4_1_1_2"/><b>then </b><i><font color="Green" title="E54">A61</font></i>: 
<font color="Maroon" title="c11">fj1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_2_4_1_1_2"><i><font color="Green" title="E52">A59</font></i></a>, <a class="txt" href="dickson.html#E2:49_1_2_4_1_1_2"><i><font color="Green" title="E53">A60</font></i></a>, <a class="ref" href="dickson.html#T9" target="_self" title="DICKSON:th.9">Th9</a></i>;<br/><b>set </b><font color="Maroon" title="c12">IC</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font></span>)</span>;<br/><a NAME="E6:49_1_2_4_1_1_2"/><i><font color="Green" title="E55">A62</font></i>: 
<font color="Maroon" title="c11">fj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:49_1_2"><i><font color="Green" title="E41">A31</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_4_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_4_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><a NAME="E2:49_1_2_4_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c10">fj</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c10">fj</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:49_1_2_4_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">fj</font> <a href="waybel_4.html#R4" title="WAYBEL_4:pred.4">is_minimal_wrt</a> <font color="Maroon" title="c5">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="dickson.html#E5:49_1_2_4_1"><i><font color="Green" title="E51">A53</font></i></a>, <a class="ref" href="dickson.html#T31" target="_self" title="DICKSON:th.31">Th31</a></i>;<br/><b>hence </b><a NAME="E4:49_1_2_4_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_2"><i><font color="Green" title="E38">A28</font></i></a>, <a class="ref" href="dickson.html#T22" target="_self" title="DICKSON:th.22">Th22</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:49_1_2_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E5:49_1_2_4_1_1_2"><i><font color="Green" title="E54">A61</font></i></a>, <a class="txt" href="dickson.html#E6:49_1_2_4_1_1_2"><i><font color="Green" title="E55">A62</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E3:49_1_2_4"/>
for <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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="dickson.html#E1:49_1_2_4"><i><font color="Green" title="E47">A49</font></i></a>, <a class="txt" href="dickson.html#E2:49_1_2_4"><i><font color="Green" title="E48">A50</font></i></a>);</i><br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">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="E14:49_1_2"/><i><font color="Green" title="E47">A63</font></i>: 
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</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="Maroon" title="c9">j</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E2:49"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/><b>thus </b><a NAME="E15:49_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E12:49_1_2"><i><font color="Green" title="E46">A48</font></i></a>, <a class="txt" href="dickson.html#E14:49_1_2"><i><font color="Green" title="E47">A63</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
