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<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> ;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E25">A1</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1"/>
( ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  &amp; not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  ) or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  ) or ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  ) or ( not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  ) or ( not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  &amp; not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:26_1_1"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>and </b><br/><a NAME="E2:26_1_1"/><i><font color="Green" title="E27">A3</font></i>: 
not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 ;<br/><div class="add"><a NAME="E3:26_1_1"/><i><font color="Green" title="E28">A4</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:26_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></i>;<br/><a NAME="E4:26_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:26_1_1"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/><a NAME="E5:26_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26_1_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="arytm_2.html#E3:26_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green" title="E21">Lm17</font></i></a></i>;<br/><b>hence </b><a NAME="E6:26_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26_1_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="arytm_2.html#E2:26_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self" title="ARYTM_2:def.5">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><b>suppose </b></a><a NAME="E1:26_1_2"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ;<br/><div class="add"><a NAME="E2:26_1_2"/><b>then </b>
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E19"><i><font color="Green" title="E13">Lm10</font></i></a></i>;<br/><a NAME="E3:26_1_2"/><b>then </b>
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><a NAME="E4:26_1_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E19"><i><font color="Green" title="E13">Lm10</font></i></a></i>;<br/><b>hence </b><a NAME="E5:26_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26_1_2"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:26_1_3"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <b>and </b><br/><a NAME="E2:26_1_3"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c5">rx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E4:26_1_3"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">rx</font>
 <b>and </b><br/><a NAME="E5:26_1_3"/><i><font color="Green" title="E29">A9</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">rx</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26_1_3"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">ry</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E7:26_1_3"/><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">ry</font>
 <b>and </b><br/><a NAME="E8:26_1_3"/><i><font color="Green" title="E31">A11</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">ry</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:26_1_3"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></i>;<br/><b>assume </b><a NAME="E9:26_1_3"/>
<font color="Maroon" title="c4">y</font> <a href="arytm_2.html#NR2" title="ARYTM_2:NR.2">&lt;</a> <font color="Maroon" title="c3">x</font>
 ;<br/><a NAME="E10:26_1_3"/><b>then </b>
<font color="Maroon" title="c6">ry</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">rx</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4:26_1_3"><i><font color="Green" title="E28">A8</font></i></a>, <a class="txt" href="arytm_2.html#E7:26_1_3"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green" title="E18">Lm14</font></i></a></i>;<br/><a NAME="E11:26_1_3"/><b>then </b>
<font color="Maroon" title="c6">ry</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:26_1_3"><i><font color="Green" title="E29">A9</font></i></a></i>;<br/><a NAME="E12:26_1_3"/><b>then </b>
<font color="Maroon" title="c6">ry</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/><a NAME="E13:26_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">ry</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c6">ry</font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E8:26_1_3"><i><font color="Green" title="E31">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E14:26_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:26_1_4"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <b>and </b><br/><a NAME="E2:26_1_4"/><i><font color="Green" title="E27">A13</font></i>: 
not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <b>and </b><br/><a NAME="E3:26_1_4"/><i><font color="Green" title="E28">A14</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c5">rx</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E5:26_1_4"/><i><font color="Green" title="E29">A15</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">rx</font>
 <b>and </b><br/><a NAME="E6:26_1_4"/><i><font color="Green" title="E30">A16</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">rx</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26_1_4"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></i>;<br/><a NAME="E7:26_1_4"/><i><font color="Green" title="E31">A17</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive" title="b2">s</font> where <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b2">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> </span>}</span>  where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:26_1_4"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="arytm_2.html#E5:26_1_4"><i><font color="Green" title="E29">A15</font></i></a>, <a class="txt" href="arytm_2.html#E6:26_1_4"><i><font color="Green" title="E30">A16</font></i></a></i>;<br/><a NAME="E8:26_1_4"/><i><font color="Green" title="E32">A18</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:26_1_4"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></i>;<br/><a NAME="E9:26_1_4"/><b>then </b>
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:26_1_4"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="arytm_2.html#E7:26_1_4"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="arytm_2.html#E29"><i><font color="Green" title="E23">Lm19</font></i></a></i>;<br/><a NAME="E10:26_1_4"/><b>then </b>
not  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="arytm_2.html#E8:26_1_4"><i><font color="Green" title="E32">A18</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">r0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E12:26_1_4"/><i><font color="Green" title="E33">A19</font></i>: 
<font color="Maroon" title="c6">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 <b>and </b><br/><a NAME="E13:26_1_4"/><i><font color="Green" title="E34">A20</font></i>: 
not <font color="Maroon" title="c6">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8:26_1_4"><i><font color="Green" title="E32">A18</font></i></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/><a NAME="E14:26_1_4"/>
<font color="Maroon" title="c5">rx</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c6">r0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:26_1_4"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="arytm_2.html#E13:26_1_4"><i><font color="Green" title="E34">A20</font></i></a></i>;<br/><a NAME="E15:26_1_4"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:26_1_4"><i><font color="Green" title="E29">A15</font></i></a>, <a class="txt" href="arytm_2.html#E8:26_1_4"><i><font color="Green" title="E32">A18</font></i></a>, <a class="txt" href="arytm_2.html#E12:26_1_4"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="arytm_2.html#E26"><i><font color="Green" title="E20">Lm16</font></i></a></i>;<br/><b>hence </b><a NAME="E16:26_1_4"/>
<font color="Maroon" title="c3">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26_1_4"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="arytm_2.html#E2:26_1_4"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self" title="ARYTM_2:def.5">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_5"><b>suppose </b></a><b>that </b><a NAME="E1:26_1_5"/><i><font color="Green" title="E26">A21</font></i>: 
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <b>and </b><br/><a NAME="E2:26_1_5"/><i><font color="Green" title="E27">A22</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <b>and </b><br/><a NAME="E3:26_1_5"/><i><font color="Green" title="E28">A23</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c5">ry</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E5:26_1_5"/><i><font color="Green" title="E29">A24</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">ry</font>
 <b>and </b><br/><a NAME="E6:26_1_5"/><i><font color="Green" title="E30">A25</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">ry</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E2:26_1_5"><i><font color="Green" title="E27">A22</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></i>;<br/><a NAME="E7:26_1_5"/><i><font color="Green" title="E31">A26</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive" title="b2">s</font> where <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b2">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> </span>}</span>  where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E3:26_1_5"><i><font color="Green" title="E28">A23</font></i></a>, <a class="txt" href="arytm_2.html#E5:26_1_5"><i><font color="Green" title="E29">A24</font></i></a>, <a class="txt" href="arytm_2.html#E6:26_1_5"><i><font color="Green" title="E30">A25</font></i></a></i>;<br/><a NAME="E8:26_1_5"/><i><font color="Green" title="E32">A27</font></i>: 
 <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26_1_5"><i><font color="Green" title="E26">A21</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></i>;<br/><a NAME="E9:26_1_5"/><b>then </b>
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive" title="b2">s</font> where <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b2">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Olive" title="b1">t</font> </span>}</span>  where <font color="Olive" title="b1">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  : <font color="Olive" title="b1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26_1_5"><i><font color="Green" title="E26">A21</font></i></a>, <a class="txt" href="arytm_2.html#E29"><i><font color="Green" title="E23">Lm19</font></i></a></i>;<br/><a NAME="E10:26_1_5"/><b>then </b>
not  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="arytm_2.html#E7:26_1_5"><i><font color="Green" title="E31">A26</font></i></a>, <a class="txt" href="arytm_2.html#E8:26_1_5"><i><font color="Green" title="E32">A27</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">r0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> <b> such that </b><br/><a NAME="E12:26_1_5"/><i><font color="Green" title="E33">A28</font></i>: 
<font color="Maroon" title="c6">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font>
 <b>and </b><br/><a NAME="E13:26_1_5"/><i><font color="Green" title="E34">A29</font></i>: 
not <font color="Maroon" title="c6">r0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8:26_1_5"><i><font color="Green" title="E32">A27</font></i></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/><a NAME="E14:26_1_5"/>
 ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r0</font> &amp; <font color="Olive" title="b1">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c5">ry</font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E6:26_1_5"><i><font color="Green" title="E30">A25</font></i></a>, <a class="txt" href="arytm_2.html#E12:26_1_5"><i><font color="Green" title="E33">A28</font></i></a></i>;<br/><a NAME="E15:26_1_5"/><b>then </b>
not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:26_1_5"><i><font color="Green" title="E29">A24</font></i></a>, <a class="txt" href="arytm_2.html#E8:26_1_5"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="arytm_2.html#E13:26_1_5"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="arytm_2.html#E26"><i><font color="Green" title="E20">Lm16</font></i></a></i>;<br/><b>hence </b><a NAME="E16:26_1_5"/>
<font color="Maroon" title="c3">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26_1_5"><i><font color="Green" title="E26">A21</font></i></a>, <a class="txt" href="arytm_2.html#E2:26_1_5"><i><font color="Green" title="E27">A22</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self" title="ARYTM_2:def.5">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_6"><b>suppose </b></a><b>that </b><a NAME="E1:26_1_6"/><i><font color="Green" title="E26">A30</font></i>: 
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 <b>and </b><br/><a NAME="E2:26_1_6"/><i><font color="Green" title="E27">A31</font></i>: 
not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 ;<br/><div class="add"><a NAME="E3:26_1_6"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_2.html#K3" title="ARYTM_2:func.3">DEDEKIND_CUT</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26_1_6"><i><font color="Green" title="E26">A30</font></i></a>, <a class="txt" href="arytm_2.html#E2:26_1_6"><i><font color="Green" title="E27">A31</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self" title="ARYTM_2:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:26_1_6"/>
<font color="Maroon" title="c3">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E1:26"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="arytm_2.html#E1:26_1_6"><i><font color="Green" title="E26">A30</font></i></a>, <a class="txt" href="arytm_2.html#E2:26_1_6"><i><font color="Green" title="E27">A31</font></i></a>, <a class="ref" href="arytm_2.html#D5" target="_self" title="ARYTM_2:def.5">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
