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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="eqrel_1.html#NM1">Tolerance</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:50"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="toler_1.html#K4" target="_self">TolClasses</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:50_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="toler_1.html#K4" target="_self">TolClasses</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="toler_1.html#K4" target="_self">TolClasses</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="toler_1.html#K4" target="_self">TolClasses</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:50_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="toler_1.html#E3:50_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="toler_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E5:50_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="toler_1.html#T43" target="_self">Th43</a></i>;<br/>
<b>hence </b><a NAME="E6:50_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="toler_1.html#E3:50_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:50_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="toler_1.html#NM2">TolClass</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:50_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="toler_1.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E5:50_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="toler_1.html#K4" target="_self">TolClasses</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="toler_1.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E6:50_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="toler_1.html#K4" target="_self">TolClasses</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="toler_1.html#E4:50_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:50"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="toler_1.html#K4" target="_self">TolClasses</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
