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<div class="add">

<b>let </b><font color="Maroon" title="c1">R</font> be  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>assume </b><a NAME="E1:76"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>
 ;<br/>

<a NAME="E2:76"/><i><font color="Green" title="E54">A2</font></i>: 
( <font color="Maroon" title="c1">R</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a> &amp; <font color="Maroon" title="c1">R</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#E1:76"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></i>;<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>thus </b><a NAME="E3:76"/>
( <font color="Maroon" title="c1">R</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a> implies for <font color="Olive" title="b1">S</font> being 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>   st <font color="Olive" title="b1">S</font> is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b1">S</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</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="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b1">S</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:76"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="dickson.html#T49" target="_self" title="DICKSON:th.49">Th49</a></i>;<br/>

<b>assume </b><a NAME="E4:76"/><i><font color="Green" title="E55">A3</font></i>: 
for <font color="Olive" title="b1">S</font> being 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>   st <font color="Olive" title="b1">S</font> is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b1">S</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</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="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b1">S</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1"/>
not <font color="Maroon" title="c1">R</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 ;<br/><a NAME="E2:76_1"/><b>then </b>
not 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="c1">R</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#E1:76"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="dickson.html#T33" target="_self" title="DICKSON:th.33">Th33</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">f</font> being   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E4:76_1"/><i><font color="Green" title="E56">A4</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">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>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/> not <font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">j</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:76"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="dickson.html#T32" target="_self" title="DICKSON:th.32">Th32</a></i>;<br/><b>defpred </b><font color="Maroon">S<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>means </b>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,$2</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="c1">R</font> or  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">j</font></span>)</span></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="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span>,$2</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="c1">R</font> ) );<br/><b>consider </b><font color="Maroon" title="c5">QOE</font> being    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,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="E6:76_1"/><i><font color="Green" title="E57">A5</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  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="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> <font color="Maroon" title="c5">QOE</font> iff ( <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="c1">R</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="c1">R</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <b>from </b><i><a class="ref" href="relset_1.html#S1" title="RELSET_1:sch.1">RELSET_1:sch 1</a>();<br/></i><b>set </b><font color="Maroon" title="c6">S</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c5">QOE</font> #);<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:76_1_1"/><i><font color="Green" title="E58">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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="c1">R</font>
 ;<br/><a NAME="E2:76_1_1"/><i><font color="Green" title="E59">A7</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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#E1:76_1_1"><i><font color="Green" title="E58">A6</font></i></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/><a NAME="E3:76_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</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#E1:76_1_1"><i><font color="Green" title="E58">A6</font></i></a>, <a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:76_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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> <font color="Maroon" title="c5">QOE</font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:76_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_1"><i><font color="Green" title="E58">A6</font></i></a>, <a class="txt" href="dickson.html#E2:76_1_1"><i><font color="Green" title="E59">A7</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:76_1"/><b>then </b><i><font color="Green" title="E58">A8</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c5">QOE</font> #)
 <b>by </b><i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></i>;<br/><a NAME="E9:76_1"/><i><font color="Green" title="E59">A9</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</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="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:76"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="orders_2.html#D4" title="ORDERS_2:def.4">ORDERS_2:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_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="E1:76_1_2"/><i><font color="Green" title="E60">A10</font></i>: 
<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/><a NAME="E2:76_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">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="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E9:76_1"><i><font color="Green" title="E59">A9</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_2"><i><font color="Green" title="E60">A10</font></i></a>, <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="E3:76_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c5">QOE</font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:76_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_2"><i><font color="Green" title="E60">A10</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E11:76_1"/><b>then </b>
<font color="Maroon" title="c5">QOE</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="c1">R</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/><a NAME="E12:76_1"/><b>then </b><i><font color="Green" title="E60">A11</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c5">QOE</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="E13:76_1"/><i><font color="Green" title="E61">A12</font></i>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</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="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:76"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:76_1_3"/><i><font color="Green" title="E62">A13</font></i>: 
( <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> &amp; <font color="Maroon" title="c8">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="c1">R</font> &amp; <font color="Maroon" title="c9">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="c1">R</font> )
 <b>and </b><br/><a NAME="E2:76_1_3"/><i><font color="Green" title="E63">A14</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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> <font color="Maroon" title="c5">QOE</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">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> <font color="Maroon" title="c5">QOE</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_3_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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="c1">R</font> or  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">j</font></span>)</span></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="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c8">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="c1">R</font> ) )
 </span><b>by </b><i><a class="txt" href="dickson.html#E6:76_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="dickson.html#E2:76_1_3"><i><font color="Green" title="E63">A14</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:76_1_3_1_1_1"/><i><font color="Green" title="E64">A15</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">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="c1">R</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_3_1_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">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="c1">R</font> or  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">j</font></span>)</span></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="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c9">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="c1">R</font> ) )
 </span><b>by </b><i><a class="txt" href="dickson.html#E6:76_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="dickson.html#E2:76_1_3"><i><font color="Green" title="E63">A14</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:76_1_3_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">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="c1">R</font>
 ;<br/><div class="add"><a NAME="E2:76_1_3_1_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">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="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E13:76_1"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_3"><i><font color="Green" title="E62">A13</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_3_1_1_1"><i><font color="Green" title="E64">A15</font></i></a>, <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="E3:76_1_3_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">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> <font color="Maroon" title="c5">QOE</font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:76_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_3"><i><font color="Green" title="E62">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:76_1_3_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">j</font></span>)</span></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="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c9">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="c1">R</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">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="E3:76_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E65">A16</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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">j</font></span>)</span></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="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<font color="Maroon" title="c9">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="c1">R</font> )
 ;<br/><a NAME="E4:76_1_3_1_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">j</font></span>)</span></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="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E13:76_1"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_3"><i><font color="Green" title="E62">A13</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_3_1_1_1"><i><font color="Green" title="E64">A15</font></i></a>, <a class="txt" href="dickson.html#E3:76_1_3_1_1_1_1_1_2"><i><font color="Green" title="E65">A16</font></i></a>, <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="E5:76_1_3_1_1_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">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> <font color="Maroon" title="c5">QOE</font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:76_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_3"><i><font color="Green" title="E62">A13</font></i></a>, <a class="txt" href="dickson.html#E3:76_1_3_1_1_1_1_1_2"><i><font color="Green" title="E65">A16</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:76_1_3_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">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> <font color="Maroon" title="c5">QOE</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:76_1_3_1_1_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">j</font></span>)</span></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="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c8">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="c1">R</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">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="E3:76_1_3_1_1_2"/><i><font color="Green" title="E64">A17</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> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">j</font></span>)</span></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="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<font color="Maroon" title="c8">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="c1">R</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_3_1_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="E1:76_1_3_1_1_2_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">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="c1">R</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span></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="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c9">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="c1">R</font> ) )
 </span><b>by </b><i><a class="txt" href="dickson.html#E6:76_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="dickson.html#E2:76_1_3"><i><font color="Green" title="E63">A14</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:76_1_3_1_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">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="c1">R</font>
 ;<br/><div class="add"><a NAME="E2:76_1_3_1_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<font color="Maroon" title="c9">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="c1">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E13:76_1"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_3"><i><font color="Green" title="E62">A13</font></i></a>, <a class="txt" href="dickson.html#E3:76_1_3_1_1_2"><i><font color="Green" title="E64">A17</font></i></a>, <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="E3:76_1_3_1_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">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> <font color="Maroon" title="c5">QOE</font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:76_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_3"><i><font color="Green" title="E62">A13</font></i></a>, <a class="txt" href="dickson.html#E3:76_1_3_1_1_2"><i><font color="Green" title="E64">A17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:76_1_3_1_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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>  st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span></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="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c9">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="c1">R</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</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="E3:76_1_3_1_1_2_1_1_2"/><i><font color="Green" title="E65">A18</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">b</font></span>)</span></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="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">a</font></span>)</span>,<font color="Maroon" title="c9">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="c1">R</font> )
 ;<br/><a NAME="E4:76_1_3_1_1_2_1_1_2"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">b</font></span>)</span></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#E13:76_1"><i><font color="Green" title="E61">A12</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_3"><i><font color="Green" title="E62">A13</font></i></a>, <a class="txt" href="dickson.html#E3:76_1_3_1_1_2"><i><font color="Green" title="E64">A17</font></i></a>, <a class="txt" href="dickson.html#E3:76_1_3_1_1_2_1_1_2"><i><font color="Green" title="E65">A18</font></i></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/><a NAME="E5:76_1_3_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</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="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E6:76_1_3_1_1_2_1_1_2"/><b>then </b>
not <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">b</font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:76_1"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/><a NAME="E7:76_1_3_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:76_1_3_1_1_2_1_1_2"><i><font color="Green" title="E65">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:76_1_3_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">j</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:76_1_3_1_1_2"><i><font color="Green" title="E64">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E9:76_1_3_1_1_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">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> <font color="Maroon" title="c5">QOE</font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:76_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_3"><i><font color="Green" title="E62">A13</font></i></a>, <a class="txt" href="dickson.html#E3:76_1_3_1_1_2"><i><font color="Green" title="E64">A17</font></i></a>, <a class="txt" href="dickson.html#E3:76_1_3_1_1_2_1_1_2"><i><font color="Green" title="E65">A18</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:76_1_3_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">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> <font color="Maroon" title="c5">QOE</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:76_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">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> <font color="Maroon" title="c5">QOE</font>
 ;<br/></div><b>end;</b></div><a NAME="E15:76_1"/><b>then </b>
<font color="Maroon" title="c5">QOE</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="c1">R</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/><a NAME="E16:76_1"/><b>then </b>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c5">QOE</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/><a NAME="E17:76_1"/><b>then </b>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c5">QOE</font> #) is <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>
 <b>by </b><i><a class="txt" href="dickson.html#E12:76_1"><i><font color="Green" title="E60">A11</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self" title="DICKSON:def.3">Def3</a></i>;<br/><a NAME="E18:76_1"/><b>then </b><i><font color="Green" title="E62">A19</font></i>: 
<a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c5">QOE</font> #) <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 <b>by </b><i><a class="txt" href="dickson.html#E4:76"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="dickson.html#E8:76_1"><i><font color="Green" title="E58">A8</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">f'</font> = <font color="Maroon" title="c4">f</font> as   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <span class="p1">(<span class="default"><a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c5">QOE</font> #) <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">n</font> be    <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> ;<br/><a NAME="E1:76_1_4"/><i><font color="Green" title="E63">A20</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E2:76_1_4"/><b>then </b>
not <font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E4:76_1"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/><a NAME="E3:76_1_4"/><b>then </b><i><font color="Green" title="E64">A21</font></i>: 
not <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>hence </b><a NAME="E4:76_1_4"/>
<font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="dickson.html#E9:76_1"><i><font color="Green" title="E59">A9</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/><a NAME="E5:76_1_4"/><i><font color="Green" title="E65">A22</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></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#E9:76_1"><i><font color="Green" title="E59">A9</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_4_1"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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> <font color="Maroon" title="c5">QOE</font>
 ;<br/><a NAME="E2:76_1_4_1"/><b>then </b>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">j</font></span>)</span></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> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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#E6:76_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="dickson.html#E3:76_1_4"><i><font color="Green" title="E64">A21</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">l</font>, <font color="Maroon" title="c10">k</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="E4:76_1_4_1"/><i><font color="Green" title="E66">A23</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">l</font>
 <b>and </b><br/><a NAME="E5:76_1_4_1"/><i><font color="Green" title="E67">A24</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">l</font></span>)</span></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> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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> )
 ;<br/><a NAME="E6:76_1_4_1"/>
( <font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">l</font> &amp; <font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="dickson.html#E5:76_1_4_1"><i><font color="Green" title="E67">A24</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="E7:76_1_4_1"/><b>then </b><i><font color="Green" title="E68">A25</font></i>: 
( <font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> &amp; <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E4:76_1"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/><a NAME="E8:76_1_4_1"/><b>then </b>
<font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/><b>hence </b><a NAME="E9:76_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E4:76_1_4_1"><i><font color="Green" title="E66">A23</font></i></a>, <a class="txt" href="dickson.html#E7:76_1_4_1"><i><font color="Green" title="E68">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><a NAME="E7:76_1_4"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></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> <font color="Maroon" title="c5">QOE</font> &amp; not <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></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> <font color="Maroon" title="c5">QOE</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a>  )
 <b>by </b><i><a class="txt" href="dickson.html#E6:76_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="dickson.html#E1:76_1_4"><i><font color="Green" title="E63">A20</font></i></a>, <a class="txt" href="dickson.html#E5:76_1_4"><i><font color="Green" title="E65">A22</font></i></a>, <a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/><a NAME="E8:76_1_4"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></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> <font color="Maroon" title="c5">QOE</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">QOE</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span>
 <b>by </b><i><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="E9:76_1_4"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></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 <span class="p1">(<span class="default"><a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c5">QOE</font> #) <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 ;<br/></div><b>end;</b></div><a NAME="E21:76_1"/><b>then </b>
<font color="Maroon" title="c7">f'</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/><b>hence </b><a NAME="E22:76_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E18:76_1"><i><font color="Green" title="E62">A19</font></i></a>, <a class="ref" href="wellfnd1.html#T15" title="WELLFND1:th.15">WELLFND1:15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:76"/>
<font color="Maroon" title="c1">R</font> is <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>
 ;<br/>


</div>
