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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/>







<a NAME="E1:15"/>
<span class="p1">{<span class="default"> <span class="p2"><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> where <font color="Olive">sb</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">tb</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">sb</font> &amp; <font color="Olive">sb</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">tb</font> &amp; <font color="Olive">tb</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="euclid.html#K1">REAL</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_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="E1:15_1"/>
<font color="Maroon">y</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">sb</font>,<font color="Olive">tb</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">sb</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">tb</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">sb</font> &amp; <font color="Olive">sb</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">tb</font> &amp; <font color="Olive">tb</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:15_1"/><b>then </b>
 ex <font color="Olive">s7</font>, <font color="Olive">t7</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">s7</font>,<font color="Olive">t7</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s7</font> &amp; <font color="Olive">s7</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t7</font> &amp; <font color="Olive">t7</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> )
 
;<br/>
<b>hence </b><a NAME="E3:15_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> 2
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:15"/>
<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> : ( <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t2</font> ) </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> 2</span>)</span>
 ;<br/>


</div>
