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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>  holds   <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Olive" title="b1">f</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font></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">  <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Maroon" title="c2">f</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font></span><br/>



<a NAME="E1:38"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</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> &amp; <font color="Olive" title="b2">B</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> &amp; not <font color="Olive" title="b1">A</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b2">B</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Olive" title="b1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</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> &amp; <font color="Maroon" title="c4">B</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> &amp; not <font color="Maroon" title="c3">A</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c4">B</font> implies <font color="Maroon" title="c4">B</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">A</font> )</span><br/>

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green" title="E32">A1</font></i>: 
( <font color="Maroon" title="c3">A</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> &amp; <font color="Maroon" title="c4">B</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c4">B</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">A</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c5">a1</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> , <font color="Maroon" title="c6">R1</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:38_1"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c6">R1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <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="c5">a1</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a> 
 <b>and </b><br/><a NAME="E4:38_1"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c6">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="taxonom1.html#D5" target="_self" title="TAXONOM1:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">a2</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> , <font color="Maroon" title="c8">R2</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:38_1"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c8">R2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <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="c7">a2</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a> 
 <b>and </b><br/><a NAME="E7:38_1"/><i><font color="Green" title="E36">A5</font></i>: 
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c8">R2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:38_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="taxonom1.html#D5" target="_self" title="TAXONOM1:def.5">Def5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_1"/>
( <font color="Maroon" title="c5">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a2</font> or <font color="Maroon" title="c5">a1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">a2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1_1"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c5">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">a2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c4">B</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">A</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <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"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">y</font> ) )</span><br/><b>assume </b><a NAME="E1:38_1_1_1_1_1"/><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">y</font> )</span><br/><b>consider </b><font color="Maroon" title="c10">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_1_1"/><i><font color="Green" title="E39">A8</font></i>: 
( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">R1</font>,<font color="Maroon" title="c10">c</font> )
 <b>by </b><i><a class="txt" href="taxonom1.html#E4:38_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="taxonom1.html#E1:38_1_1_1_1_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:38_1_1_1_1_1"/><i><font color="Green" title="E40">A9</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c8">R2</font>,<font color="Maroon" title="c10">c</font>
 ;<br/><b>take </b><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c11">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">y</font> )</span><br/><b>thus </b><a NAME="E6:38_1_1_1_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E7:38_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="taxonom1.html#E3:38_1_1_1_1_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="taxonom1.html#E5:38_1_1_1_1_1"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">y</font></span><br/><b>thus </b><a NAME="E7:38_1_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">y</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E3:38_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="taxonom1.html#E6:38_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="taxonom1.html#E1:38_1_1_1_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="taxonom1.html#E3:38_1_1_1_1_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="taxonom1.html#E5:38_1_1_1_1_1"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="taxonom1.html#E28"><i><font color="Green" title="E24">Lm2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_1_1_1_1"/>
( <font color="Maroon" title="c3">A</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c4">B</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_1_2"/><i><font color="Green" title="E37">A10</font></i>: 
<font color="Maroon" title="c5">a1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">a2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c4">B</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">A</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">y</font> be    <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"> ( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">x</font> ) )</span><br/><b>assume </b><a NAME="E1:38_1_1_1_2_1"/><i><font color="Green" title="E38">A11</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">x</font> )</span><br/><b>consider </b><font color="Maroon" title="c10">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_2_1"/><i><font color="Green" title="E39">A12</font></i>: 
( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c8">R2</font>,<font color="Maroon" title="c10">c</font> )
 <b>by </b><i><a class="txt" href="taxonom1.html#E7:38_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="taxonom1.html#E1:38_1_1_1_2_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:38_1_1_1_2_1"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c6">R1</font>,<font color="Maroon" title="c10">c</font>
 ;<br/><b>take </b><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">x</font> )</span><br/><b>thus </b><a NAME="E6:38_1_1_1_2_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E4:38_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="taxonom1.html#E3:38_1_1_1_2_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="taxonom1.html#E5:38_1_1_1_2_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">x</font></span><br/><b>thus </b><a NAME="E7:38_1_1_1_2_1"/>
<font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E3:38_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="taxonom1.html#E6:38_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="taxonom1.html#E1:38_1_1_1_2"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="taxonom1.html#E3:38_1_1_1_2_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="taxonom1.html#E5:38_1_1_1_2_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="taxonom1.html#E28"><i><font color="Green" title="E24">Lm2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_1_1_1_2"/>
( <font color="Maroon" title="c3">A</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c4">B</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">A</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:38_1"/>
( <font color="Maroon" title="c3">A</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c4">B</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c3">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:38"/>
 <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Maroon" title="c2">f</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="taxonom1.html#D1" target="_self" title="TAXONOM1:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
