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<div><span class="kw">theorem </span><a NAME="T32"><span class="comment"><font color="firebrick">:: ROUGHIF1:32</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/> (  <a href="roughif1.html#K4" title="ROUGHIF1:func.4">kappa_2</a> (<font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="roughif1.html#K1" title="ROUGHIF1:func.1">kappa</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">R</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">Y</font></span>)</span>) &amp;  <a href="roughif1.html#K1" title="ROUGHIF1:func.1">kappa</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">R</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">Y</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="roughif1.html#K1" title="ROUGHIF1:func.1">kappa</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">R</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="roughif1.html#K1" title="ROUGHIF1:func.1">kappa</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">R</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">Y</font></span>)</span>)</span>)</span> ) <span class="kw">by</span> <span class="lab"><a class="txt" href="roughif1.html#E52"><span class="lab"><font color="Green" title="E37">Lemma4f1</font></span></a>, <a class="txt" href="roughif1.html#E53"><span class="lab"><font color="Green" title="E38">Lemma4f2</font></span></a></span>;<br/></div></div>
