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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c1">R</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Olive" title="b1">X</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c2">X</font> implies <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">X</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:150"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E2:150"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E3:150"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">and </span><br/><a NAME="E4:150"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a> <font color="Maroon" title="c2">X</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="orders_1.html#D8" target="_self" title="ORDERS_1:def.8">ORDERS_1:def 8</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="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">X</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a> </span><br/>

<span class="kw">thus </span><a NAME="E5:150"/>
( <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">X</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">X</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">X</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">X</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:150"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E2:150"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E3:150"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E4:150"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="txt" href="orders_1.html#E142"><span class="lab"><font color="Green" title="E22">Lm6</font></span></a>, <a class="txt" href="orders_1.html#E143"><span class="lab"><font color="Green" title="E23">Lm7</font></span></a>, <a class="txt" href="orders_1.html#E144"><span class="lab"><font color="Green" title="E24">Lm8</font></span></a>, <a class="txt" href="orders_1.html#E145"><span class="lab"><font color="Green" title="E25">Lm9</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="orders_1.html#D5" target="_self" title="ORDERS_1:def.5">ORDERS_1:def 5</a> <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>
