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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">RD</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b3">F</font> being   <a href="filter_1.html#M2" title="FILTER_1:mode.2">BinOp</a> of <font color="Maroon" title="c1">D</font>,<font color="Olive" title="b1">RD</font>  st <font color="Olive" title="b2">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Olive" title="b3">F</font> holds <br/> <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> <font color="Olive" title="b1">RD</font>,<font color="Olive" title="b2">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Olive" title="b3">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Olive" title="b1">RD</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">F</font> being   <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>  st <font color="Olive" title="b1">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Olive" title="b2">F</font> holds <br/> <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Olive" title="b1">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Olive" title="b2">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font></span><br/><b>let </b><font color="Maroon" title="c3">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <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>  st <font color="Maroon" title="c3">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Olive" title="b1">F</font> holds <br/> <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c3">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Olive" title="b1">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font></span><br/><b>let </b><font color="Maroon" title="c4">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c4">F</font> implies  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c3">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c4">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font> )</span><br/>







<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E10">A1</font></i>: 
for <font color="Olive" title="b1">a</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="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D17" title="BINOP_1:def.17">BINOP_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c3">d</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon" title="c4">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font></span><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#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">RD</font>] <b>means </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K4" title="BINOP_1:func.4">.</a> $1,<span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c3">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<div><i><font color="Green" title="E11">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c5">a</font>]</span><br/><b>thus </b><a NAME="E1:15_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c5">a</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c3">d</font></span>)</span> = 
 <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> <font color="Maroon" title="c2">RD</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">d</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#K9" title="EQREL_1:func.9">EqClass</a> <font color="Maroon" title="c2">RD</font>,<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="filter_1.html#E1:15"><i><font color="Green" title="E10">A1</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E3:15"/>
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c2">RD</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>]
 <b>from </b><i><a class="ref" href="filter_1.html#S1" target="_self" title="FILTER_1:sch.1">FILTER_1:sch 1</a>(<a class="txt" href="filter_1.html#E2:15"><i><font color="Green" title="E11">A2</font></i></a>);</i> <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D17" title="BINOP_1:def.17">BINOP_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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