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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:86"/><i><font color="Green" title="E51">A1</font></i>: 
<font color="Maroon" title="c3">R</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E2:86"/><i><font color="Green" title="E52">A2</font></i>: 
<font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E3:86"/>
( <font color="Maroon" title="c3">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">R</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:86"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:86"/>
( <font color="Maroon" title="c3">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c3">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c3">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c3">R</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a> <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E2:86"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="orders_2.html#E182"><i><font color="Green" title="E49">Lm4</font></i></a>, <a class="ref" href="orders_1.html#T93" title="ORDERS_1:th.93">ORDERS_1:93</a>, <a class="ref" href="orders_1.html#T94" title="ORDERS_1:th.94">ORDERS_1:94</a>, <a class="ref" href="orders_1.html#T95" title="ORDERS_1:th.95">ORDERS_1:95</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a><br/>

<b>let </b><font color="Maroon" title="c4">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E5:86"/><i><font color="Green" title="E53">A3</font></i>: 
<font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 <b>and </b><br/><a NAME="E6:86"/><i><font color="Green" title="E54">A4</font></i>: 
<font color="Maroon" title="c4">Z</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> 
 ;<br/>

<a NAME="E7:86"/>
( <font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">R</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:86"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="orders_2.html#E2:86"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="orders_2.html#E5:86"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E8:86"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c3">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Z</font> )
 <b>by </b><i><a class="txt" href="orders_2.html#E6:86"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="wellord1.html#D3" title="WELLORD1:def.3">WELLORD1:def 3</a></i>;<br/>


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