<?xml version="1.0"?>
<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E27">Th31</font></span>: <a NAME="T31"><span class="comment"><font color="firebrick">:: NECKLA_3:31</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">R</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>   <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>   st  not  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <font color="Olive" title="b1">R</font> is  <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a>  holds <br/> ex <font color="Olive" title="b2">G1</font>, <font color="Olive" title="b3">G2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>   <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>   <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">G2</font> &amp;  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Olive" title="b1">R</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K2" title="NECKLA_2:func.2">sum_of</a> (<font color="Olive" title="b2">G1</font>,<font color="Olive" title="b3">G2</font>) )</div></div>
