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<b>let </b><font color="Maroon">z'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/><b>let </b><font color="Maroon">x'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/><b>let </b><font color="Maroon">y'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:65"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">z'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">x'</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E2:65"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>and </b><br/><a NAME="E3:65"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">r0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">t0</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="E5:65"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r0</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0</font>
 <b>and </b><br/><a NAME="E6:65"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">r0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E7:65"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font> )
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E24"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="arytm_3.html#T98">ARYTM_3:98</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1"/>
( <font color="Maroon">r0</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">r0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1"><b>suppose </b></a><a NAME="E1:65_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>take </b>
 <a href="arytm_3.html#K13">{}</a> 
;<br/><b>take </b>
<font color="Maroon">t0</font>
;<br/><b>thus </b><a NAME="E2:65_1_1"/>
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#K11">+</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="arytm_2.html#E26"><i><font color="Green">Lemma66</font></i></a></i>;<br/><a NAME="E3:65_1_1"/>
 <a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/><b>hence </b><a NAME="E4:65_1_1"/>
 <a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#D5" target="_self">Def5</a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/><b>thus </b><a NAME="E5:65_1_1"/>
<font color="Maroon">t0</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2"><b>suppose </b></a><a NAME="E1:65_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><a NAME="E2:65_1_2"/>
<font color="Maroon">t0</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:65_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/><b>then consider </b><font color="Maroon">t1</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="E4:65_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">t0</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t1</font>
 <b>and </b><br/><a NAME="E5:65_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">t1</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T101">ARYTM_3:101</a></i>;<br/><a NAME="E6:65_1_2"/>
<font color="Maroon">z'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t1</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E25"><i><font color="Green">Lemma63</font></i></a>, <a class="txt" href="arytm_2.html#E27"><i><font color="Green">Lemma67</font></i></a>, <a class="ref" href="arytm_3.html#T83">ARYTM_3:83</a></i>;<br/><b>then consider </b><font color="Maroon">t2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">r1</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="E8:65_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">r1</font>
 <b>and </b><br/><a NAME="E9:65_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">t2</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t1</font>
 <b>and </b><br/><a NAME="E10:65_1_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">r1</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r0</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r0</font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E26"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="arytm_3.html#T98">ARYTM_3:98</a></i>;<br/><b>take </b>
<font color="Maroon">r1</font>
;<br/><b>take </b>
<font color="Maroon">t2</font>
;<br/><b>thus </b><a NAME="E11:65_1_2"/>
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t2</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E29"><i><font color="Green">Lemma69</font></i></a></i>;<br/><a NAME="E12:65_1_2"/>
<font color="Maroon">r1</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r0</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/><b>hence </b><a NAME="E13:65_1_2"/>
<font color="Maroon">r1</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">x'</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/><b>thus </b><a NAME="E14:65_1_2"/>
<font color="Maroon">t2</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E28"><i><font color="Green">Lemma68</font></i></a>, <a class="txt" href="arytm_2.html#E30"><i><font color="Green">Lemma70</font></i></a>, <a class="ref" href="arytm_3.html#T76">ARYTM_3:76</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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