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<div><span class="kw">theorem </span><a NAME="T19"><span class="comment"><font color="firebrick">:: ROUGHIF2:19</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">R</font> being   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="roughs_1.html#NM1" title="ROUGHS_1:NM.1">Approximation_Space</a><br/>  for <font color="Olive" title="b2">X</font>, <font color="Olive" title="b3">Y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">R</font> holds <br/> ( <span class="p1">(<span class="default"><a href="roughif2.html#K1" title="ROUGHIF2:func.1">CMap</a> <span class="p2">(<span class="default"><a href="roughif1.html#K6" title="ROUGHIF1:func.6">kappa_2</a> <font color="Olive" title="b1">R</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="roughif2.html#K1" title="ROUGHIF2:func.1">CMap</a> <span class="p2">(<span class="default"><a href="roughif1.html#K5" title="ROUGHIF1:func.5">kappa_1</a> <font color="Olive" title="b1">R</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</font>) &amp; <span class="p1">(<span class="default"><a href="roughif2.html#K1" title="ROUGHIF2:func.1">CMap</a> <span class="p2">(<span class="default"><a href="roughif1.html#K5" title="ROUGHIF1:func.5">kappa_1</a> <font color="Olive" title="b1">R</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="roughif2.html#K1" title="ROUGHIF2:func.1">CMap</a> <span class="p2">(<span class="default"><a href="roughif1.html#K2" title="ROUGHIF1:func.2">kappa</a> <font color="Olive" title="b1">R</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</font>) ) <span class="kw">by</span> <span class="lab"><a class="txt" href="roughif2.html#E23"><span class="lab"><font color="Green" title="E16">Lemma6c1</font></span></a>, <a class="txt" href="roughif2.html#E24"><span class="lab"><font color="Green" title="E17">Lemma6c2</font></span></a></span>;<br/></div></div>
