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<div><span class="kw">theorem </span><a NAME="T5"><span class="comment"><font color="firebrick">:: REALALG3:5</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">R</font> being   <a href="vectsp_2.html#NM2" title="VECTSP_2:NM.2">domRing</a><br/>  for <font color="Olive" title="b2">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">R</font>  st <font color="Olive" title="b2">S</font> is  <a href="realalg1.html#V12" title="REALALG1:attr.12">positive_cone</a>  holds <br/>( <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">S</font> <a href="realalg3.html#K4" title="REALALG3:func.4">^+</a></span>)</span>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">R</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">S</font> <a href="realalg3.html#K5" title="REALALG3:func.5">^-</a></span>)</span></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> &amp; <font color="Olive" title="b2">S</font> <a href="realalg3.html#K4" title="REALALG3:func.4">^+</a>  is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>  &amp; <font color="Olive" title="b2">S</font> <a href="realalg3.html#K4" title="REALALG3:func.4">^+</a>  is  <a href="realalg1.html#V5" title="REALALG1:attr.5">mult-closed</a>  )</div></div>
