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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">a</font> being non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">a</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">f</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a> holds <br/> <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Olive" title="b1">f</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">a</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">f</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a> holds <br/> <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Maroon" title="c2">f</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/><b>let </b><font color="Maroon" title="c3">a</font> be  non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">a</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">f</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a> implies  <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Maroon" title="c2">f</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>





<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E30">A1</font></i>: 
(  <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">a</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">f</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Maroon" title="c2">f</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<b>reconsider </b><font color="Maroon" title="c4">R</font> = <span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="taxonom1.html#E1:36"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="taxonom1.html#T22" target="_self" title="TAXONOM1:th.22">Th22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> =  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">R</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E4:36"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#D5" target="_self" title="TAXONOM1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E5:36"/>
 <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Maroon" title="c2">f</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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