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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>  &amp; <font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a>  implies <font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:52"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a> 
 <span class="kw">and </span><br/><a NAME="E2:52"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V4" title="DICKSON:attr.4">Dickson</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> </span><br/>

<a NAME="E3:52"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="dickson.html#T30" target="_self" title="DICKSON:th.30">Th30</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  ex <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="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c2">N</font> )</span><br/><span class="kw">assume </span><a NAME="E1:52_1"/>
<font color="Maroon" title="c2">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c2">N</font></span><br/><a NAME="E2:52_1"/><span class="kw">then </span>
not  <a href="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c2">N</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="dickson.html#T32" target="_self" title="DICKSON:th.32">Th32</a></span>;<br/><span class="kw">hence </span><a NAME="E3:52_1"/>
 ex <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="dickson.html#K6" title="DICKSON:func.6">min-classes</a> <font color="Maroon" title="c2">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:52"/>
<font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dickson.html#T21" target="_self" title="DICKSON:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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