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<b>let </b><font color="Maroon">s1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">t1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">t2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> or  not <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> or  not <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span> 
 ;<br/><a NAME="E2:18_1"/><b>then </b>
 ex <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> or  not <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> or  not <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) )
 ;<br/><a NAME="E3:18_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span>  )
 ;<br/><a NAME="E4:18_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:18_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:18"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> or  not <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> or  not <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span> 
 ;<br/><a NAME="E2:18_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:18_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:18_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:18_2"/><b>then </b>
(  ex <font color="Olive">sa</font>, <font color="Olive">ta</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">sa</font>,<font color="Olive">ta</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">sa</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> ) or  ex <font color="Olive">sc</font>, <font color="Olive">tc</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">sc</font>,<font color="Olive">tc</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">tc</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> ) or  ex <font color="Olive">sb</font>, <font color="Olive">tb</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">sb</font>,<font color="Olive">tb</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">sb</font> ) or  ex <font color="Olive">sd</font>, <font color="Olive">td</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">sd</font>,<font color="Olive">td</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">td</font> ) )
 ;<br/><b>hence </b><a NAME="E6:18_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> or  not <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> or  not <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span> 
 ;<br/></div><b>end;</b></div>
<a NAME="E4:18"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> or  not <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> or  not <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:18"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> or  not <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> or  not <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s3</font>,<font color="Olive">t3</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s3</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t3</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p3">{<span class="default"> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s4</font>,<font color="Olive">t4</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s4</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t4</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">t4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t1</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s5</font>,<font color="Olive">t5</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t5</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s6</font>,<font color="Olive">t6</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s6</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t6</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">t2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t6</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
