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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a> <a href="roughs_1.html#NM1" title="ROUGHS_1:NM.1">Approximation_Space</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>







<b>set </b><font color="Maroon" title="c5">CI</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">x</font>;<br/>
<b>assume </b><a NAME="E1:77"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<a NAME="E2:77"/><i><font color="Green" title="E38">A2</font></i>: 
<span class="p1">(<span class="default"><a href="roughs_1.html#K6" title="ROUGHS_1:func.6">MemberFunc</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="roughs_1.html#D9" target="_self" title="ROUGHS_1:def.9">Def9</a></i>;<br/>
<a NAME="E3:77"/>
<font color="Maroon" title="c2">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="roughs_1.html#E1:77"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E4:77"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T80" title="CARD_1:th.80">CARD_1:80</a></i>;<br/>
<a NAME="E5:77"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/>
<b>hence </b><a NAME="E6:77"/>
<span class="p1">(<span class="default"><a href="roughs_1.html#K6" title="ROUGHS_1:func.6">MemberFunc</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K6" title="ROUGHS_1:func.6">MemberFunc</a> <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="roughs_1.html#E2:77"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="roughs_1.html#D9" target="_self" title="ROUGHS_1:def.9">Def9</a></i>;<br/>


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