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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="roughs_1.html#NM2" title="ROUGHS_1:NM.2">Tolerance_Space</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font> holds   <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p2">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Olive" title="b1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Olive" title="b1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p2">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span><br/>



<span class="kw">thus </span><a NAME="E1:51"/>
 <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p2">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p2">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p2">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p2">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E2:51_1"/><span class="kw">then </span>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">x</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_1.html#T10" target="_self" onmouseover="rs('roughs_1/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:51_1"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">x</font>)
 <span class="kw">and </span><br/><a NAME="E5:51_1"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<a NAME="E6:51_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="eqrel_1.html#T19" onmouseover="rs('eqrel_1/T19')" onmouseout="rh()">EQREL_1:19</a></span>;<br/>
<a NAME="E7:51_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T6" onmouseover="rs('eqrel_1/T6')" onmouseout="rh()">EQREL_1:6</a></span>;<br/>
<a NAME="E8:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T19" onmouseover="rs('eqrel_1/T19')" onmouseout="rh()">EQREL_1:19</a></span>;<br/>
<a NAME="E9:51_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">y</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="roughs_1.html#T8" target="_self" onmouseover="rs('roughs_1/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:51_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E2:51"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_2"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span></span><br/>

<a NAME="E2:51_2"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">x</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">x</font>) or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_2_1"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">x</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E2:51_2_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T19" onmouseover="rs('eqrel_1/T19')" onmouseout="rh()">EQREL_1:19</a></span>;<br/>
<a NAME="E3:51_2_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T6" onmouseover="rs('eqrel_1/T6')" onmouseout="rh()">EQREL_1:6</a></span>;<br/>
<a NAME="E4:51_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T19" onmouseover="rs('eqrel_1/T19')" onmouseout="rh()">EQREL_1:19</a></span>;<br/>
<a NAME="E5:51_2_1"/><span class="kw">then </span>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> ( the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c4">y</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:51_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2"><span class="lab"><font color="Green" title="E25">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:51"/>
 <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <span class="p1">(<span class="default"><a href="roughs_1.html#K3" title="ROUGHS_1:func.3">LAp</a> <span class="p2">(<span class="default"><a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_1.html#T25" target="_self" onmouseover="rs('roughs_1/T25')" onmouseout="rh()">Th25</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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