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<b>let </b><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="arytm_2.html#K2" target="_self">REAL+</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) or ( not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) or ( not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:27_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E2:27_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E1:27_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="arytm_2.html#E2:27_1_1"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E4:27_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="arytm_2.html#E2:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma19</font></i></a></i>;<br/><b>hence </b><a NAME="E5:27_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T73">ARYTM_3:73</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:27_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E2:27_1_2"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><a NAME="E3:27_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="arytm_2.html#E2:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="arytm_2.html#E1:27_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="arytm_2.html#E2:27_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E4:27_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:27_1_3"/><i><font color="Green">E29</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E2:27_1_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><a NAME="E3:27_1_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="arytm_2.html#E2:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="arytm_2.html#E1:27_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="arytm_2.html#E2:27_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E4:27_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:27_1_4"/><i><font color="Green">E29</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E2:27_1_4"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><a NAME="E3:27_1_4"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:27"><i><font color="Green">E27</font></i></a>, <a class="txt" href="arytm_2.html#E2:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="arytm_2.html#E1:27_1_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="arytm_2.html#E2:27_1_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E4:27_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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