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<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  &amp; <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">z</font> implies <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> )</span><br/>





<b>assume </b><a NAME="E1:71"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">z</font> or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c6">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> <b> such that </b><br/><a NAME="E3:71"/><i><font color="Green" title="E65">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="arytm_3.html#T60" target="_self" title="ARYTM_3:th.60">Th60</a></i>;<br/>
<a NAME="E4:71"/>
( <font color="Maroon" title="c6">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c6">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="arytm_3.html#K12" title="ARYTM_3:func.12">one</a>  <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">z</font> )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E3:71"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="arytm_3.html#T58" target="_self" title="ARYTM_3:th.58">Th58</a></i>;<br/>
<a NAME="E5:71"/><b>then </b>
( <font color="Maroon" title="c6">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c6">r</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T59" target="_self" title="ARYTM_3:th.59">Th59</a></i>;<br/>
<b>hence </b><a NAME="E6:71"/>
(  not <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="arytm_3.html#K10" title="ARYTM_3:func.10">*'</a> <font color="Maroon" title="c5">z</font> or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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