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<b>let </b><font color="Maroon" title="c1">D</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> ;<br/><b>let </b><font color="Maroon" title="c2">RD</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be    <a href="filter_1.html#M2" title="FILTER_1:mode.2">BinOp</a> of <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">RD</font>;<br/>





<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">RD</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">RD</font>] <b>means </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> $1,$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> $2,$1;<br/>
<div><i><font color="Green" title="E8">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/><b>thus </b><a NAME="E1:12_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="eqrel_1.html#K10" title="EQREL_1:func.10">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c4">x1</font>, <a href="eqrel_1.html#K10" title="EQREL_1:func.10">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c5">x2</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K10" title="EQREL_1:func.10">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c4">x1</font></span>)</span>,<span class="p1">(<span class="default"><a href="eqrel_1.html#K10" title="EQREL_1:func.10">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c5">x2</font></span>)</span> = 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c2">RD</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="filter_1.html#T3" target="_self" title="FILTER_1:th.3">Th3</a></i>
<br/>.= 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <font color="Maroon" title="c2">RD</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c4">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="filter_1.html#E1:12"><i><font color="Green" title="E7">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K10" title="EQREL_1:func.10">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p1">(<span class="default"><a href="eqrel_1.html#K10" title="EQREL_1:func.10">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c4">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="filter_1.html#T3" target="_self" title="FILTER_1:th.3">Th3</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E3:12"/>
for <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c2">RD</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c1</font>,<font color="Olive" title="b2">c2</font>]
 <b>from </b><i><a class="ref" href="filter_1.html#S2" target="_self" title="FILTER_1:sch.2">FILTER_1:sch 2</a>(<a class="txt" href="filter_1.html#E2:12"><i><font color="Green" title="E8">A2</font></i></a>);</i> <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a><br/>


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