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<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> ;<br/>





<b>thus </b><a NAME="E1:8"/>
( <font color="Maroon" title="c2">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon" title="c2">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">z0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T10" title="ARYTM_2:th.10">ARYTM_2:10</a></i>;<br/>
<a NAME="E4:8_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font>
 
<b>by </b><i><a class="txt" href="arytm_1.html#E3:8_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="arytm_2.html#T7" title="ARYTM_2:th.7">ARYTM_2:7</a></i>;<br/>
<b>hence </b><a NAME="E5:8_1"/>
<font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:8"/>
<font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">z0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_2.html#K2" title="ARYTM_2:func.2">REAL+</a> <b> such that </b><br/><a NAME="E4:8"/><i><font color="Green" title="E8">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T10" title="ARYTM_2:th.10">ARYTM_2:10</a></i>;<br/>
<a NAME="E5:8"/>
<font color="Maroon" title="c3">y</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c5">z0</font></span>)</span> <a href="arytm_2.html#K7" title="ARYTM_2:func.7">+</a> <font color="Maroon" title="c4">z</font>
 
<b>by </b><i><a class="txt" href="arytm_1.html#E4:8"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="arytm_2.html#T7" title="ARYTM_2:th.7">ARYTM_2:7</a></i>;<br/>
<b>hence </b><a NAME="E6:8"/>
<font color="Maroon" title="c2">x</font> <a href="arytm_2.html#R1" title="ARYTM_2:pred.1">&lt;='</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T12" title="ARYTM_2:th.12">ARYTM_2:12</a>, <a class="ref" href="arytm_2.html#T20" title="ARYTM_2:th.20">ARYTM_2:20</a></i>;<br/>


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