<?xml version="1.0"?>
<div class="add">

<span class="kw">assume </span><a NAME="E1:5"/>
 <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:5"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c3">e</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 class="kw">and </span><br/><a NAME="E4:5"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c3">e</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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">t</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> <span class="kw"> such that </span><br/><a NAME="E6:5"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c3">e</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="c4">t</font> </span>}</span> 
 <span class="kw">and </span><br/><a NAME="E7:5"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>;<br/>
<a NAME="E8:5"/>
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c4">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T71" title="ARYTM_3:th.71">ARYTM_3:71</a></span>;<br/>
<a NAME="E9:5"/><span class="kw">then </span>
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="arytm_3.html#T75" title="ARYTM_3:th.75">ARYTM_3:75</a></span>;<br/>
<a NAME="E10:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A5</font></span>: 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:5"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>;<br/>
<a NAME="E11:5"/>
<font color="Maroon" title="c3">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">e</font> or <font color="Maroon" title="c5">u</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/>

<span class="kw">assume </span><a NAME="E1:5_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">u</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/>

<a NAME="E2:5_1"/><span class="kw">then </span>
 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="c5">u</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="c4">t</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:5"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:5_1"/>
<font color="Maroon" title="c5">u</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c3">e</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:5"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">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> <span class="kw"> such that </span><br/><a NAME="E14:5"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">and </span><br/><a NAME="E15:5"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
for <font color="Olive" title="b1">r</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 <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">e</font> holds <br/><font color="Olive" title="b1">r</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c6">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="arytm_3.html#T99" title="ARYTM_3:th.99">ARYTM_3:99</a></span>;<br/>
<a NAME="E16:5"/>
 ex <font color="Olive" title="b1">r</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">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Olive" title="b1">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">t</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:5"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E8">A6</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">r</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> <span class="kw"> such that </span><br/><a NAME="E18:5"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">and </span><br/><a NAME="E19:5"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T101" title="ARYTM_3:th.101">ARYTM_3:101</a></span>;<br/>
<a NAME="E20:5"/>
<font color="Maroon" title="c7">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:5"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E19:5"><span class="lab"><font color="Green" title="E11">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:5"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E18:5"><span class="lab"><font color="Green" title="E10">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
