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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> iff  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> ) )</span><br/>



<span class="kw">set </span><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/>
<span class="kw">thus </span><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> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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> ) implies <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:50_1"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:50_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:50_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">Y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/>

<span class="kw">thus </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E6:50_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="coh_sp.html#D16" target="_self" title="COH_SP:def.16">Def19</a></span>; <a class="txt" 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">given </span><font color="Maroon" title="c4">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E3:50"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:50"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E29">A4</font></span></a></span>;<br/>
<a NAME="E6:50"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <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> 
 
;<br/>
<a NAME="E7:50"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="coh_sp.html#D16" target="_self" title="COH_SP:def.16">Def19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:50"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" 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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