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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="realset2.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2">suppf</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:9"/>
(  not <span class="p1">(<span class="default"><a href="realset2.html#K5">omf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E9</font></i>: 
<span class="p1">(<span class="default"><a href="realset2.html#K5">omf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:9_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:9_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:9_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="realset2.html#K2">suppf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:9_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="realset2.html#K8">revf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="realset2.html#K2">suppf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="ref" href="realset2.html#T28">REALSET2:28</a></i>;<br/>
<a NAME="E6:9_1"/><b>then </b><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="realset2.html#K8">revf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2">suppf</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="realset2.html#K5">omf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="realset2.html#K6">nmf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="realset2.html#T25">REALSET2:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K5">omf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K5">omf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="realset2.html#K8">revf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="realset3.html#E4:9_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="realset2.html#D15">REALSET2:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K5">omf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K5">omf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K8">revf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="realset3.html#E6:9_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="realset2.html#T23">REALSET2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="realset2.html#K5">omf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K8">revf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="realset3.html#E1:9_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="realset2.html#T24">REALSET2:24</a></i>
<br/>.= 
 <a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="realset3.html#E6:9_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="realset2.html#T18">REALSET2:18</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:9"/>
( <span class="p1">(<span class="default"><a href="realset2.html#K5">omf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4">ndf</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="realset2.html#T17">REALSET2:17</a>, <a class="ref" href="realset2.html#T18">REALSET2:18</a></i>;<br/>


</div>
