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

<a NAME="E1:14"/>
(  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E4:14_1"/>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 
<b>by </b><i><a class="txt" href="realset2.html#E3:14_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:14_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="realset2.html#E3:14_1"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:14"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> iff ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> ) )
 <b>by </b><i><a class="txt" href="realset2.html#E13"><i><font color="Green">Lemma10</font></i></a>, <a class="txt" href="realset2.html#E14"><i><font color="Green">Lemma11</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


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