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<b>let </b><font color="Maroon">A</font> be  <a href="group_1.html#V6">finite</a> <a href="roughs_1.html#NM1">Approximation_Space</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/>



<b>thus </b><a NAME="E1:89"/>
 <a href="roughs_1.html#K5" target="_self">BndAp</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K5" target="_self">BndAp</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E3:89_1"/><i><font color="Green">E27</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:89_1"/>
<span class="p1">(<span class="default"><a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:89_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="roughs_1.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:89"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">y'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:89"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> &amp; <span class="p1">(<span class="default"><a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/>
<b>thus </b><a NAME="E5:89"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K5" target="_self">BndAp</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div>
