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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V6">finite</a> <a href="orders_3.html#V1">discrete</a> <a href="roughs_1.html#NM1">Approximation_Space</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:74"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E4:74"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:74_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K3" target="_self">LAp</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="roughs_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E3:74_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="roughs_1.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E4:74_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="roughs_1.html#E1:74_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_2"/><i><font color="Green">E37</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:74_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="roughs_1.html#K4" target="_self">UAp</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="roughs_1.html#T16" target="_self">Th16</a></i>;<br/><a NAME="E3:74_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K4" target="_self">UAp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="roughs_1.html#E1:74_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="kurato_2.html#T1">KURATO_2:1</a></i>;<br/><a NAME="E4:74_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="roughs_1.html#T41" target="_self">Th41</a></i>;<br/><b>hence </b><a NAME="E5:74_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="roughs_1.html#E1:74_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="roughs_1.html#E1:74_1_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:74"/>
 <a href="roughs_1.html#K6" target="_self">MemberFunc</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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