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<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1"/>
( ex <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 <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>  holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) or <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#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  &amp; ( 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>  holds <br/>not 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>  holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>suppose </b></a><a NAME="E1:16_1_1"/>
ex <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 <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>  holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E3:16_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:16_1_1"/><i><font color="Green">E18</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>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self">Def4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E6:16_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E7:16_1_1"/><i><font color="Green">E20</font></i>: 
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where 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="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:16_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self">Def3</a></i>;<br/><b>thus </b><a NAME="E8:16_1_1"/>
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:16_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:16_1_1_1"/><b>then </b>
ex <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 <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</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="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="arytm_2.html#E7:16_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_1_1"/>
<font color="Maroon">c<sub>6</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#E4:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="arytm_2.html#E6:16_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">c<sub>6</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="E9:16_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  ;<br/><a NAME="E11:16_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="arytm_2.html#E9:16_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><b>hence </b><a NAME="E12:16_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:16_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="arytm_2.html#E7:16_1_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><b>suppose </b></a><a NAME="E1:16_1_2"/><i><font color="Green">E17</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"><a NAME="E2:16_1_2"/><b>then </b>
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</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#E20"><i><font color="Green">Lemma15</font></i></a></i>;<br/><b>hence </b><a NAME="E3:16_1_2"/>
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:16_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma14</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:16_1_3"/><i><font color="Green">E17</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="E2:16_1_3"/><i><font color="Green">E18</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>  holds <br/>not 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>  holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E3:16_1_3"/><i><font color="Green">E19</font></i>: 
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</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_2.html#E2:16_1_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="arytm_2.html#D4" target="_self">Def4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_3_1"/>
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><a NAME="E2:16_1_3_1"/><b>then </b>
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <a href="arytm_3.html#K7">RAT+</a>  <a href="subset_1.html#K9">/\</a> <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:16_1_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:16_1_3_1"/><b>then </b>
 <a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</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#E18"><i><font color="Green">Lemma13</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:16_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:16_1_3"><i><font color="Green">E17</font></i></a>, <a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma15</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:16_1_3"/>
 <a href="arytm_2.html#K3" target="_self">DEDEKIND_CUT</a> <span class="p1">(<span class="default"><a href="arytm_2.html#K4" target="_self">GLUED</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:16_1_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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