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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:15"/><i><font color="Green">E18</font></i>: 
<span class="p1"><a href="arytm_0.html#K5" target="_self">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="arytm_0.html#K5" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5" target="_self">[*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="arytm_0.html#K5" target="_self">*]</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>suppose </b></a><a NAME="E1:15_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:15_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="arytm_0.html#K5" target="_self">[*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="arytm_0.html#K5" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="arytm_0.html#D7" target="_self">Def7</a></i>;<br/><div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:15_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="arytm_0.html#E2:15_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="arytm_0.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="arytm_0.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:15_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="arytm_0.html#E2:15_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="arytm_0.html#D7" target="_self">Def7</a></i>;<br/><b>thus </b><a NAME="E5:15_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:15_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="arytm_0.html#E3:15_1_1"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2"><b>suppose </b></a><a NAME="E1:15_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:15_1_2"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="arytm_0.html#K5" target="_self">[*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="arytm_0.html#K5" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="arytm_0.html#E1:15"><i><font color="Green">E18</font></i></a>, <a class="ref" href="arytm_0.html#D7" target="_self">Def7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:15_1_2_1"/><b>then </b>
<span class="p1"><a href="arytm_0.html#K5" target="_self">[*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="arytm_0.html#K5" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="arytm_0.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E3:15_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:15_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="arytm_0.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div><a NAME="E4:15_1_2"/><b>then </b>
<span class="p1"><a href="arytm_0.html#K5" target="_self">[*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="arytm_0.html#K5" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> 0,1 <a href="funct_4.html#K5">--&gt;</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="arytm_0.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E5:15_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="arytm_0.html#E2:15_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_4.html#T72">FUNCT_4:72</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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