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<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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font>, <font color="Olive" title="b3">Z</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">X</font> <a href="roughs_1.html#R2" title="ROUGHS_1:pred.2">c=^</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b2">Y</font> <a href="roughs_1.html#R2" title="ROUGHS_1:pred.2">c=^</a> <font color="Olive" title="b3">Z</font> holds <br/><font color="Olive" title="b1">X</font> <a href="roughs_1.html#R2" title="ROUGHS_1:pred.2">c=^</a> <font color="Olive" title="b3">Z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font>, <font color="Maroon" title="c4">Z</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R2" title="ROUGHS_1:pred.2">c=^</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="roughs_1.html#R2" title="ROUGHS_1:pred.2">c=^</a> <font color="Maroon" title="c4">Z</font> implies <font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R2" title="ROUGHS_1:pred.2">c=^</a> <font color="Maroon" title="c4">Z</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:91"/>
( <font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R2" title="ROUGHS_1:pred.2">c=^</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c3">Y</font> <a href="roughs_1.html#R2" title="ROUGHS_1:pred.2">c=^</a> <font color="Maroon" title="c4">Z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R2" title="ROUGHS_1:pred.2">c=^</a> <font color="Maroon" title="c4">Z</font></span><br/>

<a NAME="E2:91"/><span class="kw">then </span>
(  <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> <font color="Maroon" title="c3">Y</font> &amp;  <a href="roughs_1.html#K4" title="ROUGHS_1:func.4">UAp</a> <font color="Maroon" title="c3">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="c4">Z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_1.html#D12" target="_self" title="ROUGHS_1:def.12">Def12</a></span>;<br/>
<a NAME="E3:91"/><span class="kw">then </span>
 <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> <font color="Maroon" title="c4">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:91"/>
<font color="Maroon" title="c2">X</font> <a href="roughs_1.html#R2" title="ROUGHS_1:pred.2">c=^</a> <font color="Maroon" title="c4">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_1.html#D12" target="_self" title="ROUGHS_1:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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