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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">T</font> be   <a href="eqrel_1.html#NM1">Tolerance</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font> &amp;  ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Z</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> ) );<br/>
<b>consider </b><font color="Maroon">TS</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:34"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:34"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; <font color="Olive">x</font> is    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:34_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font> implies ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> is    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font>
 ;<br/>

<b>hence </b><a NAME="E2:34_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> is    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="toler_1.html#T8" target="_self">Th8</a></i>;<br/>

<a NAME="E3:34_1_1"/>
 ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">Z</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> )
 
<b>by </b><i><a class="ref" href="toler_1.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>hence </b><a NAME="E4:34_1_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:34_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> is    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font> )
 <b>by </b><i><a class="ref" href="toler_1.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:34"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">TS</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<a NAME="E1:34_2"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="toler_1.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E2:34_2"/>
<font color="Maroon">TS</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="toler_1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:34"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">TS</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font>
 ;<br/>

<b>hence </b><a NAME="E2:34_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="toler_1.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:34"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">TS</font> &amp; <font color="Olive">Z</font> is <a href="ordinal1.html#V6">c=-linear</a> holds <br/> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font> &amp; ( for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> holds <br/><font color="Olive">X1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_4"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">TS</font> &amp; <font color="Maroon">Z</font> is <a href="ordinal1.html#V6">c=-linear</a> )
 ;<br/>

<a NAME="E2:34_4"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:34_4_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">Zx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_4_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Zx</font> &amp; <font color="Maroon">Zx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">Zy</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:34_4_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Zy</font> &amp; <font color="Maroon">Zy</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Zx</font> = <font color="Maroon">Zx</font> as    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="toler_1.html#T8" target="_self">Th8</a>, <a class="txt" href="toler_1.html#E3:34"><i><font color="Green">E22</font></i></a>, <a class="ref" href="toler_1.html#T27" target="_self">Th27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Zy</font> = <font color="Maroon">Zy</font> as    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="toler_1.html#T8" target="_self">Th8</a>, <a class="txt" href="toler_1.html#E3:34"><i><font color="Green">E22</font></i></a>, <a class="ref" href="toler_1.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E8:34_4_1"/>
<font color="Maroon">Zx</font>,<font color="Maroon">Zy</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i><a class="txt" href="toler_1.html#E3:34"><i><font color="Green">E22</font></i></a>, <a class="ref" href="toler_1.html#T27" target="_self">Th27</a>, <a class="ref" href="toler_1.html#T32" target="_self">Th32</a>, <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E9:34_4_1"/><b>then </b>
( <font color="Maroon">Zx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Zy</font> or <font color="Maroon">Zy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Zx</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<b>hence </b><a NAME="E10:34_4_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="toler_1.html#T27" target="_self">Th27</a>, <a class="ref" href="toler_1.html#T32" target="_self">Th32</a>, <a class="ref" href="toler_1.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:34_4"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font> &amp; ( for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">X1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_4_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">Y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">TS</font>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E3:34_4_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E4:34_4_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>hence </b><a NAME="E5:34_4_2"/>
<font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="toler_1.html#T27" target="_self">Th27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:34_4"/>
( <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font> &amp; ( for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">X1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_4_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>take </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
;<br/>

<a NAME="E2:34_4_3"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="toler_1.html#E2:34"><i><font color="Green">E21</font></i></a>, <a class="txt" href="toler_1.html#E3:34"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:34_4_3"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E4:34_4_3"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T99">ZFMISC_1:99</a></i>;<br/>
<a NAME="E5:34_4_3"/><i><font color="Green">E48</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> is    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="toler_1.html#T15" target="_self">Th15</a>, <a class="ref" href="toler_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E6:34_4_3"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_4_3_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/>
<a NAME="E3:34_4_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font>
 
<b>by </b><i><a class="txt" href="toler_1.html#E3:34"><i><font color="Green">E22</font></i></a>, <a class="ref" href="toler_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:34_4_3_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="toler_1.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E5:34_4_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="toler_1.html#T27" target="_self">Th27</a>, <a class="txt" href="toler_1.html#E4:34"><i><font color="Green">E28</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="E7:34_4_3"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font>
 <b>by </b><i><a class="ref" href="toler_1.html#T9" target="_self">Th9</a>, <a class="ref" href="toler_1.html#T32" target="_self">Th32</a>, </i>;<br/>

<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E8:34_4_3"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>hence </b><a NAME="E9:34_4_3"/>
<font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:34_4"/>
 ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font> &amp; ( for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/><font color="Olive">X1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">Z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:34"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font> &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Z1</font> holds <br/> not <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> ) )
 <b>by </b><i>, <a class="ref" href="orders_1.html#T175">ORDERS_1:175</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Z1</font> = <font color="Maroon">Z1</font> as    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="toler_1.html#T8" target="_self">Th8</a>, <a class="txt" href="toler_1.html#E3:34"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E10:34"/>
<font color="Maroon">Z1</font> is   <a href="toler_1.html#NM2">TolClass</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_5"/>
<font color="Maroon">Z1</font> is  not   <a href="toler_1.html#NM2">TolClass</a> of <font color="Maroon">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_5"/><i><font color="Green">E58</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> ) )
 <b>by </b><i><a class="ref" href="toler_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>set </b><font color="Maroon">Y1</font> = <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E4:34_5"/>
for <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_5_1"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:34_5_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/>

<a NAME="E2:34_5_1"/><b>then </b>
( ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) &amp; ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:34_5_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:34_5_1"/><i><font color="Green">E60</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E5:34_5_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/>
<b>hence </b><a NAME="E6:34_5_1"/>
contradiction
 <b>by </b><i><a class="ref" href="toler_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="toler_1.html#T27" target="_self">Th27</a>, <a class="ref" href="toler_1.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:34_5"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> is    <a href="toler_1.html#M1" target="_self">TolSet</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="toler_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E6:34_5"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="toler_1.html#T8" target="_self">Th8</a>, <a class="txt" href="toler_1.html#E3:34"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E7:34_5"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="toler_1.html#T15" target="_self">Th15</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:34_5"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E9:34_5"/>
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> &amp; <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="toler_1.html#T9" target="_self">Th9</a>, <a class="txt" href="toler_1.html#E3:34"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:34_5"/><b>then </b>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:34_5"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">TS</font>
 
<b>by </b><i><a class="ref" href="toler_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="toler_1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E12:34_5"/>
<font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_5_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">Z1</font>
 ;<br/>

<a NAME="E2:34_5_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:34_5_2"/>
contradiction
 <b>by </b><i><a class="ref" href="toler_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:34_5"/><b>then </b>
not <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="toler_1.html#E3:34"><i><font color="Green">E22</font></i></a>, <a class="ref" href="toler_1.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E14:34_5"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Z1</font> = <font color="Maroon">Z1</font> as   <a href="toler_1.html#NM2">TolClass</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">Z1</font>
;<br/>

<b>thus </b><a NAME="E12:34"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font>
 <b>by </b><i><a class="ref" href="toler_1.html#T9" target="_self">Th9</a>, <a class="txt" href="toler_1.html#E3:34"><i><font color="Green">E22</font></i></a></i>;<br/>


</div>
