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<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">RD</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="filter_1.html#M2" target="_self">BinOp</a> of <font color="Maroon">D</font>,<font color="Maroon">RD</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="filter_1.html#M2" target="_self">BinOp</a> of <font color="Maroon">D</font>,<font color="Maroon">RD</font>;<br/>







<b>assume </b><a NAME="E1:17"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">d</font>, <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d</font>,<font color="Olive">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Olive">d</font>,<font color="Olive">b</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D18">BINOP_1:def 18</a><br/>

<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">RD</font> =  <a href="eqrel_1.html#K10">EqClass</a> <font color="Maroon">RD</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">RD</font>,   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">RD</font>,   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">RD</font>] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>3</sub></font></span>)</span>;<br/>
<div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>, <font color="Maroon">x2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>, <font color="Maroon">x3</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>thus </b><a NAME="E1:17_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="eqrel_1.html#K10">EqClass</a> <font color="Maroon">RD</font>,<font color="Maroon">x1</font>, <a href="eqrel_1.html#K10">EqClass</a> <font color="Maroon">RD</font>,<font color="Maroon">x2</font>, <a href="eqrel_1.html#K10">EqClass</a> <font color="Maroon">RD</font>,<font color="Maroon">x3</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">x1</font>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">x2</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">x3</font>)</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">x1</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x2</font>,<font color="Maroon">x3</font>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x2</font>,<font color="Maroon">x3</font></span>)</span>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x3</font></span>)</span>)
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x3</font>)
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">x1</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">x2</font>)</span>)</span>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x3</font>)
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">x1</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">x2</font>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">RD</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">x1</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">x3</font>)</span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E3:17"/>
for <font color="Olive">c1</font>, <font color="Olive">c2</font>, <font color="Olive">c3</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">RD</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c1</font>,<font color="Olive">c2</font>,<font color="Olive">c3</font>]
 <b>from </b><i><a class="ref" href="filter_1.html#S3" target="_self">FILTER_1:sch 3</a>();</i> <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D18">BINOP_1:def 18</a><br/>


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