<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> ;<br/>





<b>thus </b><a NAME="E1:8"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T10">ARYTM_2:10</a></i>;<br/>
<a NAME="E4:8_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_1.html#E3:8_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="arytm_2.html#T7">ARYTM_2:7</a></i>;<br/>
<b>hence </b><a NAME="E5:8_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T20">ARYTM_2:20</a></i>;<br/>


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

<b>assume </b><a NAME="E2:8"/>
<font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2">REAL+</a> <b> such that </b><br/><a NAME="E4:8"/><i><font color="Green">E8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T10">ARYTM_2:10</a></i>;<br/>
<a NAME="E5:8"/>
<font color="Maroon">c<sub>3</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_1.html#E4:8"><i><font color="Green">E8</font></i></a>, <a class="ref" href="arytm_2.html#T7">ARYTM_2:7</a></i>;<br/>
<a NAME="E6:8"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="arytm_2.html#T12">ARYTM_2:12</a></i>;<br/>
<b>hence </b><a NAME="E7:8"/>
<font color="Maroon">c<sub>2</sub></font> <a href="arytm_2.html#R1">&lt;='</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T20">ARYTM_2:20</a></i>;<br/>


</div>
