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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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="E1:44"/><i><font color="Green">E42</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="E2:44"/><i><font color="Green">E43</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>5</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_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>  &amp; <font color="Maroon">c<sub>5</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; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; not <font color="Maroon">c<sub>5</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>  &amp; <font color="Maroon">c<sub>5</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>  &amp; not <font color="Maroon">c<sub>5</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>  &amp; <font color="Maroon">c<sub>5</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>  &amp; not <font color="Maroon">c<sub>5</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>  &amp; <font color="Maroon">c<sub>5</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>  &amp; not <font color="Maroon">c<sub>5</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="44_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:44_1_1"/><i><font color="Green">E44</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="E2:44_1_1"/><i><font color="Green">E45</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> 
 <b>and </b><br/><a NAME="E3:44_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</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>6</sub></font> = <font color="Maroon">c<sub>3</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#E1:44_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</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#E2:44_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</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#E3:44_1_1"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E7:44_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44"><i><font color="Green">E42</font></i></a>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma19</font></i></a></i>;<br/><a NAME="E8:44_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44"><i><font color="Green">E43</font></i></a>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma19</font></i></a></i>;<br/><a NAME="E9:44_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E7:44_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_3.html#T74">ARYTM_3:74</a></i>;<br/><b>hence </b><a NAME="E10:44_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:44_1_2"/><i><font color="Green">E44</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="E2:44_1_2"/><i><font color="Green">E45</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> 
 <b>and </b><br/><a NAME="E3:44_1_2"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">c<sub>5</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>6</sub></font> = <font color="Maroon">c<sub>3</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#E1:44_1_2"><i><font color="Green">E44</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</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#E2:44_1_2"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E6:44_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E7:44_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:44_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E9:44_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E10:44_1_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><a NAME="E11:44_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44"><i><font color="Green">E42</font></i></a>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma19</font></i></a></i>;<br/><a NAME="E12:44_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44"><i><font color="Green">E43</font></i></a>, <a class="txt" href="arytm_2.html#E2:44_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E3:44_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E9:44_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E13:44_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E10:44_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="arytm_2.html#E11:44_1_2"><i><font color="Green">E49</font></i></a></i>;<br/><b>hence </b><a NAME="E14:44_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:44_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E9:44_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:44_1_3"/><i><font color="Green">E44</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="E2:44_1_3"/><i><font color="Green">E45</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> 
 <b>and </b><br/><a NAME="E3:44_1_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</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>6</sub></font> = <font color="Maroon">c<sub>3</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#E1:44_1_3"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E5:44_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E6:44_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E8:44_1_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E9:44_1_3"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</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#E3:44_1_3"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E11:44_1_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44"><i><font color="Green">E42</font></i></a>, <a class="txt" href="arytm_2.html#E2:44_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E8:44_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E12:44_1_3"/>
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44"><i><font color="Green">E43</font></i></a>, <a class="txt" href="arytm_2.html#E2:44_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E8:44_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E13:44_1_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E9:44_1_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="arytm_2.html#E11:44_1_3"><i><font color="Green">E49</font></i></a></i>;<br/><b>hence </b><a NAME="E14:44_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:44_1_4"/><i><font color="Green">E44</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="E2:44_1_4"/><i><font color="Green">E45</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> 
 <b>and </b><br/><a NAME="E3:44_1_4"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">c<sub>5</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>6</sub></font> = <font color="Maroon">c<sub>3</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#E1:44_1_4"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E5:44_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E6:44_1_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44_1_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E8:44_1_4"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E9:44_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><a NAME="E10:44_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E11:44_1_4"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:44_1_4"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E13:44_1_4"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E14:44_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><a NAME="E15:44_1_4"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44"><i><font color="Green">E42</font></i></a>, <a class="txt" href="arytm_2.html#E2:44_1_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E8:44_1_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E16:44_1_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44"><i><font color="Green">E43</font></i></a>, <a class="txt" href="arytm_2.html#E2:44_1_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E3:44_1_4"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E8:44_1_4"><i><font color="Green">E47</font></i></a>, <a class="txt" href="arytm_2.html#E13:44_1_4"><i><font color="Green">E48</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E17:44_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:44_1_4"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E13:44_1_4"><i><font color="Green">E48</font></i></a>, <a class="txt" href="arytm_2.html#E15:44_1_4"><i><font color="Green">E49</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_5"><b>suppose </b></a><b>that </b><a NAME="E1:44_1_5"/><i><font color="Green">E44</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="E2:44_1_5"/><i><font color="Green">E45</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> 
 <b>and </b><br/><a NAME="E3:44_1_5"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><a NAME="E4:44_1_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E5:44_1_5"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44_1_5"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E7:44_1_5"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E8:44_1_5"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</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#E2:44_1_5"><i><font color="Green">E45</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</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#E3:44_1_5"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E11:44_1_5"/><i><font color="Green">E49</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44"><i><font color="Green">E42</font></i></a>, <a class="txt" href="arytm_2.html#E1:44_1_5"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E7:44_1_5"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E12:44_1_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44"><i><font color="Green">E43</font></i></a>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma19</font></i></a></i>;<br/><a NAME="E13:44_1_5"/><b>then </b>
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8:44_1_5"><i><font color="Green">E48</font></i></a>, <a class="txt" href="arytm_2.html#E11:44_1_5"><i><font color="Green">E49</font></i></a></i>;<br/><b>hence </b><a NAME="E14:44_1_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44_1_5"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E7:44_1_5"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_6"><b>suppose </b></a><b>that </b><a NAME="E1:44_1_6"/><i><font color="Green">E44</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="E2:44_1_6"/><i><font color="Green">E45</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> 
 <b>and </b><br/><a NAME="E3:44_1_6"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><a NAME="E4:44_1_6"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E5:44_1_6"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44_1_6"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E7:44_1_6"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E8:44_1_6"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</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#E2:44_1_6"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E10:44_1_6"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E11:44_1_6"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:44_1_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E13:44_1_6"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E14:44_1_6"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><a NAME="E15:44_1_6"/><i><font color="Green">E51</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44"><i><font color="Green">E42</font></i></a>, <a class="txt" href="arytm_2.html#E1:44_1_6"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E7:44_1_6"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E16:44_1_6"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44"><i><font color="Green">E43</font></i></a>, <a class="txt" href="arytm_2.html#E2:44_1_6"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E3:44_1_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E13:44_1_6"><i><font color="Green">E49</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E17:44_1_6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:44_1_6_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  ;<br/>
<a NAME="E3:44_1_6_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E8:44_1_6"><i><font color="Green">E48</font></i></a>, <a class="txt" href="arytm_2.html#E15:44_1_6"><i><font color="Green">E51</font></i></a>, <a class="txt" href="arytm_2.html#E1:44_1_6_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:44_1_6_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E14:44_1_6"><i><font color="Green">E50</font></i></a>, <a class="txt" href="arytm_2.html#E16:44_1_6"><i><font color="Green">E52</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E18:44_1_6"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44_1_6"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E3:44_1_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E7:44_1_6"><i><font color="Green">E47</font></i></a>, <a class="txt" href="arytm_2.html#E13:44_1_6"><i><font color="Green">E49</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_7"><b>suppose </b></a><b>that </b><a NAME="E1:44_1_7"/><i><font color="Green">E44</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="E2:44_1_7"/><i><font color="Green">E45</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> 
 <b>and </b><br/><a NAME="E3:44_1_7"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><a NAME="E4:44_1_7"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E5:44_1_7"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44_1_7"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E7:44_1_7"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E8:44_1_7"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><a NAME="E9:44_1_7"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E10:44_1_7"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44_1_7"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E12:44_1_7"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E13:44_1_7"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</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#E3:44_1_7"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E15:44_1_7"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44"><i><font color="Green">E42</font></i></a>, <a class="txt" href="arytm_2.html#E1:44_1_7"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E2:44_1_7"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E7:44_1_7"><i><font color="Green">E47</font></i></a>, <a class="txt" href="arytm_2.html#E12:44_1_7"><i><font color="Green">E48</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E16:44_1_7"/><b>then </b>
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44"><i><font color="Green">E43</font></i></a>, <a class="txt" href="arytm_2.html#E2:44_1_7"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E12:44_1_7"><i><font color="Green">E48</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E17:44_1_7"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44_1_7"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E7:44_1_7"><i><font color="Green">E47</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_8"><b>suppose </b></a><b>that </b><a NAME="E1:44_1_8"/><i><font color="Green">E44</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="E2:44_1_8"/><i><font color="Green">E45</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> 
 <b>and </b><br/><a NAME="E3:44_1_8"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><a NAME="E4:44_1_8"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E5:44_1_8"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44_1_8"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E7:44_1_8"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E8:44_1_8"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><a NAME="E9:44_1_8"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E10:44_1_8"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44_1_8"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E12:44_1_8"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E13:44_1_8"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><a NAME="E14:44_1_8"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E15:44_1_8"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>3</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:44_1_8"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma5</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E17:44_1_8"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E18:44_1_8"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/>( ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/><a NAME="E19:44_1_8"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44"><i><font color="Green">E42</font></i></a>, <a class="txt" href="arytm_2.html#E1:44_1_8"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E2:44_1_8"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E7:44_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="arytm_2.html#E12:44_1_8"><i><font color="Green">E48</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E20:44_1_8"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:44"><i><font color="Green">E43</font></i></a>, <a class="txt" href="arytm_2.html#E2:44_1_8"><i><font color="Green">E45</font></i></a>, <a class="txt" href="arytm_2.html#E3:44_1_8"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E12:44_1_8"><i><font color="Green">E48</font></i></a>, <a class="txt" href="arytm_2.html#E17:44_1_8"><i><font color="Green">E49</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E21:44_1_8"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E19:44_1_8"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E22:44_1_8"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:44_1_8"><i><font color="Green">E44</font></i></a>, <a class="txt" href="arytm_2.html#E3:44_1_8"><i><font color="Green">E46</font></i></a>, <a class="txt" href="arytm_2.html#E7:44_1_8"><i><font color="Green">E47</font></i></a>, <a class="txt" href="arytm_2.html#E17:44_1_8"><i><font color="Green">E49</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
