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<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<a NAME="E1:21"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="arytm_2.html#K2">REAL+</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K13">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="numbers.html#D1">NUMBERS:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/><div class="add"><a NAME="E2:21_1_1"/><b>then </b>
 ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> &amp;  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">y'</font> )
 <b>by </b><i><a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:21_1_1"/>
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<a href="arytm_2.html#K2">REAL+</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:21_1_2"/><b>then </b>
 ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">x'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Olive">y'</font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font> <a href="arytm_2.html#K8">*'</a> <font color="Olive">x'</font> )
 <b>by </b><i><a class="txt" href="arytm_0.html#E1"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>hence </b><a NAME="E3:21_1_2"/>
 <a href="arytm_0.html#K2" target="_self">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2">REAL+</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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