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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">f</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="coh_sp.html#K15" title="COH_SP:func.15">Toler</a> <font color="Olive" title="b1">Y</font></span>)</span> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> : verum </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:50"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K16" title="COH_SP:func.16">Toler_on_subsets</a> <font color="Maroon" title="c2">X</font> implies  ex <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c1">x</font> is   <a href="eqrel_1.html#NM1" title="EQREL_1:NM.1">Tolerance</a> of <font color="Olive" title="b1">A</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K16" title="COH_SP:func.16">Toler_on_subsets</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E2:50_1"/><b>then </b>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="coh_sp.html#K15" title="COH_SP:func.15">Toler</a> <font color="Olive" title="b1">Y</font></span>)</span> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> : verum </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c4">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:50_1"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="coh_sp.html#K15" title="COH_SP:func.15">Toler</a> <font color="Olive" title="b1">Y</font></span>)</span> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">Y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E6:50_1"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K15" title="COH_SP:func.15">Toler</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="coh_sp.html#E4:50_1"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">Y</font>
;<br/>

<b>thus </b><a NAME="E7:50_1"/>
( <font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c1">x</font> is   <a href="eqrel_1.html#NM1" title="EQREL_1:NM.1">Tolerance</a> of <font color="Maroon" title="c5">Y</font> )
 <b>by </b><i><a class="txt" href="coh_sp.html#E4:50_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="coh_sp.html#E6:50_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="coh_sp.html#D19" target="_self" title="COH_SP:def.19">Def19</a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c4">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:50"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c1">x</font> is   <a href="eqrel_1.html#NM1" title="EQREL_1:NM.1">Tolerance</a> of <font color="Maroon" title="c4">A</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">A</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="coh_sp.html#E2:50"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/>
<a NAME="E4:50"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K15" title="COH_SP:func.15">Toler</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="coh_sp.html#K15" title="COH_SP:func.15">Toler</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="coh_sp.html#K15" title="COH_SP:func.15">Toler</a> <font color="Olive" title="b1">Y</font></span>)</span> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> : verum </span>}</span>  )
 
<b>by </b><i><a class="txt" href="coh_sp.html#E2:50"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="coh_sp.html#D19" target="_self" title="COH_SP:def.19">Def19</a></i>;<br/>
<a NAME="E5:50"/><b>then </b>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="coh_sp.html#K15" title="COH_SP:func.15">Toler</a> <font color="Olive" title="b1">Y</font></span>)</span> where <font color="Olive" title="b1">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:50"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K16" title="COH_SP:func.16">Toler_on_subsets</a> <font color="Maroon" title="c2">X</font>
 ;<br/>


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