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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="firebrick">::  thesis: </font></span></a><span class="hide"> (  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> &amp;  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  &amp;  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  implies ( <font color="Maroon" title="c1">r</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  &amp; <font color="Maroon" title="c1">r</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  &amp; <font color="Maroon" title="c1">r</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">i</font> =  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">c</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">r</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
 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>
 <span class="kw">and </span><br/><a NAME="E2:23"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
(  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  &amp;  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">r</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  &amp; <font color="Maroon" title="c1">r</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  &amp; <font color="Maroon" title="c1">r</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  )</span><br/>

<a NAME="E3:23"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 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>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></span>;<br/>
<a NAME="E4:23"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</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="Maroon" title="c1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="partit_2.html#T21" title="PARTIT_2:th.21">PARTIT_2:21</a></span>;<br/>
<a NAME="E5:23"/><span class="kw">then </span>
 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#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:23"/>
( <font color="Maroon" title="c1">r</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  &amp; <font color="Maroon" title="c1">r</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  &amp; <font color="Maroon" title="c1">r</font> is  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="orders_2.html#D2" title="ORDERS_2:def.2">ORDERS_2:def 2</a>, <a class="ref" href="orders_2.html#D3" title="ORDERS_2:def.3">ORDERS_2:def 3</a>, <a class="ref" href="orders_2.html#D4" title="ORDERS_2:def.4">ORDERS_2:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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