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<b>let </b><font color="Maroon" title="c1">O</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">O</font> is <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a> implies <font color="Maroon" title="c1">O</font> is <a href="oposet_1.html#V4" title="OPOSET_1:attr.4">SubIrreFlexive</a> )</span><br/>

<b>set </b><font color="Maroon" title="c2">RO</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">O</font>;<br/>
<b>assume </b><a NAME="E1:40"/>
<font color="Maroon" title="c1">O</font> is <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</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">O</font> is <a href="oposet_1.html#V4" title="OPOSET_1:attr.4">SubIrreFlexive</a></span><br/>

<a NAME="E2:40"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">O</font> <a href="relat_2.html#R2" title="RELAT_2:pred.2">is_irreflexive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">O</font>
 
<b>by </b><i><a class="ref" href="oposet_1.html#D12" target="_self" title="OPOSET_1:def.12">Def12</a></i>;<br/>
<a NAME="E3:40"/><b>then </b>
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">O</font> is <a href="relat_2.html#V2" title="RELAT_2:attr.2">irreflexive</a>
 
<b>by </b><i><a class="ref" href="oposet_1.html#T17" target="_self" title="OPOSET_1:th.17">Th17</a></i>;<br/>
<b>hence </b><a NAME="E4:40"/>
<font color="Maroon" title="c1">O</font> is <a href="oposet_1.html#V4" title="OPOSET_1:attr.4">SubIrreFlexive</a>
 <b>by </b><i><a class="ref" href="oposet_1.html#D11" target="_self" title="OPOSET_1:def.11">Def11</a></i>; <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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