<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <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>  ex <font color="Olive" title="b2">R</font> being  <a href="alg_1.html#NM2" title="ALG_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K3" title="TAXONOM1:func.3">dist_toler</a> <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b1">a</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp;  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<b>consider </b><font color="Maroon" title="c2">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:44_1_1"/><i><font color="Green" title="E36">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:44_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:44_1_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:44_1_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E2:44_1_1"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c3">X1</font> = <font color="Maroon" title="c2">X</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">X1</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E5:44_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X1</font> implies  ex <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>  ex <font color="Olive" title="b2">R</font> being  <a href="alg_1.html#NM2" title="ALG_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K3" title="TAXONOM1:func.3">dist_toler</a> <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b1">a</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp;  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> ) )
 <b>by </b><i><a class="txt" href="taxonom1.html#E2:44_1_1"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>

<b>given </b><font color="Maroon" title="c5">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> , <font color="Maroon" title="c6">R</font> being   <a href="alg_1.html#NM2" title="ALG_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">M</font><b> such that </b><a NAME="E6:44_1_1"/><i><font color="Green" title="E37">A2</font></i>: 
( <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K3" title="TAXONOM1:func.3">dist_toler</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp;  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 ;<br/>

<a NAME="E7:44_1_1"/>
 <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c6">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K1" title="PARTIT1:func.1">PARTITIONS</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="partit1.html#D3" title="PARTIT1:def.3">PARTIT1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:44_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X1</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E2:44_1_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="taxonom1.html#E6:44_1_1"><i><font color="Green" title="E37">A2</font></i></a></i>;<br/>


</div>
