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<span class="kw">let </span><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</font>, <font color="Maroon" title="c5">r</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"> ( <font color="Maroon" title="c3">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font> iff <font color="Maroon" title="c3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:90"/>
( <font color="Maroon" title="c3">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font> implies <font color="Maroon" title="c3">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</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">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font> implies <font color="Maroon" title="c3">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c6">z</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><a NAME="E2:90_1"/><span class="lab"><font color="Green" title="E77">A1</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">z</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">ARYTM_3:def 13</a> <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">s</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">z</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">ARYTM_3:def 13</a> <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="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">z</font></span><br/>

<a NAME="E3:90_1"/>
<font color="Maroon" title="c5">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90_1"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="arytm_3.html#T51" target="_self" title="ARYTM_3:th.51">Th57</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:90_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T62" target="_self" title="ARYTM_3:th.62">Th69</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/>


</div><span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c6">z</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><a NAME="E3:90"/><span class="lab"><font color="Green" title="E77">A2</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">z</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">ARYTM_3:def 13</a> <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">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font> <a href="arytm_3.html#R3" title="ARYTM_3:pred.3">&lt;='</a> <font color="Maroon" title="c5">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">z</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="arytm_3.html#D13" target="_self" title="ARYTM_3:def.13">ARYTM_3:def 13</a> <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="c5">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">z</font></span><br/>

<span class="kw">thus </span><a NAME="E4:90"/>
<font color="Maroon" title="c5">r</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="ref" href="arytm_3.html#T51" target="_self" title="ARYTM_3:th.51">Th57</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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