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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">R</font>, <font color="Maroon" title="c3">T</font> be   <a href="eqrel_1.html#NM1" title="EQREL_1:NM.1">Tolerance</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>thus </b><a NAME="E1:55"/>
(  <a href="toler_1.html#K3" title="TOLER_1:func.3">TolSets</a> <font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="toler_1.html#K3" title="TOLER_1:func.3">TolSets</a> <font color="Maroon" title="c3">T</font> implies <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c3">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1"/><i><font color="Green" title="E24">A1</font></i>: 
 <a href="toler_1.html#K3" title="TOLER_1:func.3">TolSets</a> <font color="Maroon" title="c2">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="toler_1.html#K3" title="TOLER_1:func.3">TolSets</a> <font color="Maroon" title="c3">T</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a><br/><b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E2:55_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">Z</font> being   <a href="toler_1.html#NM2" title="TOLER_1:NM.2">TolClass</a> of <font color="Maroon" title="c2">R</font><b> such that </b><br/><a NAME="E4:55_1"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font> )
 <b>by </b><i><a class="ref" href="toler_1.html#T46" target="_self" title="TOLER_1:th.46">Th46</a></i>;<br/>
<a NAME="E5:55_1"/>
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toler_1.html#K3" title="TOLER_1:func.3">TolSets</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="toler_1.html#D6" target="_self" title="TOLER_1:def.6">Def6</a></i>;<br/>
<a NAME="E6:55_1"/><b>then </b>
<font color="Maroon" title="c6">Z</font> is    <a href="toler_1.html#M1" title="TOLER_1:mode.1">TolSet</a> of <font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="txt" href="toler_1.html#E1:55_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="toler_1.html#D6" target="_self" title="TOLER_1:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E7:55_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="toler_1.html#E4:55_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="toler_1.html#D3" target="_self" title="TOLER_1:def.3">Def3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:55"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c3">T</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:55"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toler_1.html#K3" title="TOLER_1:func.3">TolSets</a> <font color="Maroon" title="c2">R</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">Z</font> = <font color="Maroon" title="c4">x</font> as    <a href="toler_1.html#M1" title="TOLER_1:mode.1">TolSet</a> of <font color="Maroon" title="c2">R</font> <b>by </b><i><a class="ref" href="toler_1.html#D6" target="_self" title="TOLER_1:def.6">Def6</a></i>;<br/>
<a NAME="E5:55"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:55_2"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font> )
 ;<br/>

<a NAME="E2:55_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="toler_1.html#D3" target="_self" title="TOLER_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E3:55_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="toler_1.html#E2:55"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:55"/><b>then </b>
<font color="Maroon" title="c5">Z</font> is    <a href="toler_1.html#M1" title="TOLER_1:mode.1">TolSet</a> of <font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="ref" href="toler_1.html#D3" target="_self" title="TOLER_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E7:55"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toler_1.html#K3" title="TOLER_1:func.3">TolSets</a> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="ref" href="toler_1.html#D6" target="_self" title="TOLER_1:def.6">Def6</a></i>;<br/>


</div>
