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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>



<b>assume </b><a NAME="E1:53"/>
( <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> &amp; <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> &amp; <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> &amp; <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> )
 ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>hence </b><a NAME="E2:53"/>
( <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> )
 <b>by </b><i><a class="txt" href="orders_1.html#E142"><i><font color="Green" title="E25">Lm6</font></i></a>, <a class="txt" href="orders_1.html#E143"><i><font color="Green" title="E26">Lm7</font></i></a>, <a class="txt" href="orders_1.html#E144"><i><font color="Green" title="E27">Lm8</font></i></a>, <a class="txt" href="orders_1.html#E145"><i><font color="Green" title="E28">Lm9</font></i></a></i>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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