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<span class="kw">let </span><font color="Maroon" title="c3">r1</font>, <font color="Maroon" title="c4">r2</font>, <font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s2</font> or <font color="Maroon" title="c3">r1</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">s1</font> or <font color="Maroon" title="c4">r2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c6">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:97"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
<font color="Maroon" title="c3">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c4">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r1</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">s1</font> or <font color="Maroon" title="c4">r2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c6">s2</font> )</span><br/>

<a NAME="E2:97"/><span class="lab"><font color="Green" title="E80">A2</font></span>: 
 <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="c5">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T64" target="_self" title="ARYTM_3:th.64">Th71</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:97"/><span class="lab"><font color="Green" title="E81">A3</font></span>: 
<font color="Maroon" title="c5">s1</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c3">r1</font>
 <span class="kw">and </span><br/><a NAME="E4:97"/><span class="lab"><font color="Green" title="E82">A4</font></span>: 
<font color="Maroon" title="c6">s2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c4">r2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:97"/>
( <font color="Maroon" title="c6">s2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">r2</font> &amp; <font color="Maroon" title="c5">s1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s2</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c3">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="ref" href="arytm_3.html#T82" target="_self" title="ARYTM_3:th.82">Th90</a></span>;<br/>
<a NAME="E6:97"/><span class="kw">then </span>
<font color="Maroon" title="c5">s1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s2</font> <a href="arytm_3.html#NR4" title="ARYTM_3:NR.4">&lt;</a> <font color="Maroon" title="c3">r1</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="ref" href="arytm_3.html#T56" target="_self" title="ARYTM_3:th.56">Th62</a>, <a class="ref" href="arytm_3.html#T66" target="_self" title="ARYTM_3:th.66">Th73</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:97"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="ref" href="arytm_3.html#T82" target="_self" title="ARYTM_3:th.82">Th90</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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