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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">a</font> being non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ex <font color="Olive">R</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,<font color="Olive">a</font></span>)</span> <a href="lang1.html#K17">[*]</a>  &amp;  <a href="eqrel_1.html#K9">Class</a> <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>let </b><font color="Maroon">X1</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">X2</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:35_1_2"/><i><font color="Green">E31</font></i>: 
( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff  ex <font color="Olive">a</font> being non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ex <font color="Olive">R</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,<font color="Olive">a</font></span>)</span> <a href="lang1.html#K17">[*]</a>  &amp;  <a href="eqrel_1.html#K9">Class</a> <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) ) ) &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> iff  ex <font color="Olive">a</font> being non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ex <font color="Olive">R</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,<font color="Olive">a</font></span>)</span> <a href="lang1.html#K17">[*]</a>  &amp;  <a href="eqrel_1.html#K9">Class</a> <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) ) ) )
 ;<br/>

<a NAME="E2:35_1_2"/>
for <font color="Olive">X1</font>, <font color="Olive">X2</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="partit1.html#K1">PARTITIONS</a> <font color="Maroon">X</font></span>)</span>  st ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) ) &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X2</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) ) holds <br/><font color="Olive">X1</font> <a href="hidden.html#R1">=</a> <font color="Olive">X2</font>
 
<b>from </b><i><a class="ref" href="lmod_7.html#S1">LMOD_7:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E3:35_1_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font>
 <b>by </b><i/>;<br/>


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