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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><a NAME="E1:50"/><i><font color="Green">E49</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>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>suppose </b></a><a NAME="E1:50_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/><b>thus </b><a NAME="E2:50_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">c<sub>4</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_2.html#E1:50_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="arytm_2.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><b>suppose </b></a><a NAME="E1:50_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:50_1_2"/>
 <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="arytm_3.html#K13">{}</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  <b>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self">Th1</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/><b>thus </b><a NAME="E4:50_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#K7" target="_self">+</a> <font color="Maroon">c<sub>5</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_2.html#E1:50_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="arytm_2.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:50_1_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>and </b><br/><a NAME="E2:50_1_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E3:50_1_3"/><i><font color="Green">E52</font></i>: 
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:50_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma14</font></i></a></i>;<br/><a NAME="E4:50_1_3"/><i><font color="Green">E53</font></i>: 
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:50_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="arytm_2.html#E32"><i><font color="Green">Lemma27</font></i></a></i>;<br/><a NAME="E5:50_1_3"/><i><font color="Green">E54</font></i>: 
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:50"><i><font color="Green">E49</font></i></a>, <a class="txt" href="arytm_2.html#E57"><i><font color="Green">Lemma47</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> <b> such that </b><br/><a NAME="E7:50_1_3"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="arytm_2.html#K5" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:50_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="arytm_2.html#E4:50_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="arytm_2.html#E56"><i><font color="Green">Lemma46</font></i></a></i>;<br/><b>take </b>
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>5</sub></font>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1_3_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><a NAME="E2:50_1_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#K11">+</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_3_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:50_1_3_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#K11">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:50_1_3_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <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="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#K11">+</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E1:50_1_3_1_1"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:50_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:50_1_3_1"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:50_1_3_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#K11">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , B is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:50_1_3_1"/><b>then </b>
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E7:50_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="arytm_2.html#D6" target="_self">Def6</a></i>;<br/><b>hence </b><a NAME="E5:50_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:50_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="arytm_2.html#E5:50_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><a NAME="E9:50_1_3"/><b>then </b>
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma15</font></i></a></i>;<br/><b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#K7" target="_self">+</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="arytm_2.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <span class="p3">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E1:50_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="arytm_2.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="arytm_2.html#E7:50_1_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="arytm_2.html#E21"><i><font color="Green">Lemma16</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="arytm_2.html#E33"><i><font color="Green">Lemma28</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
