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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="filter_1.html#M2" target="_self">BinOp</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D19">BINOP_1:def 19</a><br/>

<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font> =  <a href="eqrel_1.html#K10">EqClass</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font>,   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font>,   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font>] means <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></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>,<font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></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>;<br/>
<div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E1:18_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="eqrel_1.html#K10">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>, <a href="eqrel_1.html#K10">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>, <a href="eqrel_1.html#K10">EqClass</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>5</sub></font>),<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>6</sub></font>)</span>)</span>,<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>) = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>),<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>)
<b>by </b><i><a class="ref" href="filter_1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>)
<b>by </b><i><a class="ref" href="filter_1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>)
<b>by </b><i><a class="txt" href="filter_1.html#E1:18"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>),<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>)
<b>by </b><i><a class="ref" href="filter_1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>5</sub></font>),<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>)</span>)</span>,<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>)
<b>by </b><i><a class="ref" href="filter_1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>5</sub></font>),<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="filter_1.html#K3" target="_self">/\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>6</sub></font>),<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>)</span>)</span>
<b>by </b><i><a class="ref" href="filter_1.html#T3" target="_self">Th3</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E3:18"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 <b>from </b><i><a class="ref" href="filter_1.html#S3" target="_self">FILTER_1:sch 3</a>(<a class="txt" href="filter_1.html#E2:18"><i><font color="Green">E13</font></i></a>);</i> <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D19">BINOP_1:def 19</a><br/>


</div>
