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

<b>let </b><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">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="c3">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c4">e</font>, <font color="Maroon" title="c5">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font>;<br/><b>let </b><font color="Maroon" title="c6">F</font>, <font color="Maroon" title="c7">G</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font>;<br/><b>let </b><font color="Maroon" title="c8">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>;<br/>















<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c6">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c6">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 <b>and </b><br/><a NAME="E3:14"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c6">F</font>
 <b>and </b><br/><a NAME="E4:14"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c6">F</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font>] <b>means </b><font color="Maroon" title="c7">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">[;]</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">f</font></span>)</span>;<br/>
<b>assume </b><a NAME="E5:14"/>
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font>
 ;<br/>

<b>then </b><font color="Maroon" title="c7">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font></span>)</span>,<font color="Maroon" title="c8">f</font></span>)</span> = 
<font color="Maroon" title="c4">e</font>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="setwop_2.html#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="setwop_2.html#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">[;]</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="setwop_2.html#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="setwop_2.html#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></i>
;<br/>
<a NAME="E7:14"/><b>then </b><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c1">C</font>]
 
;<br/>
<a NAME="E8:14"/><i><font color="Green" title="E14">A6</font></i>: 
for <font color="Olive" title="b1">B'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font><br/> for <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">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B'</font>] &amp; not <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B'</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B'</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">B'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c10">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:14_2"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B'</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">[;]</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:14_2"/><i><font color="Green" title="E16">A8</font></i>: 
not <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B'</font>
 ;<br/>

<b>thus </b><font color="Maroon" title="c7">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">B'</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c8">f</font></span>)</span> = 
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B'</font>,<font color="Maroon" title="c8">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="setwop_2.html#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="setwop_2.html#E2:14_2"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="setwop_2.html#T4" target="_self" title="SETWOP_2:th.4">Th4</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B'</font>,<font color="Maroon" title="c8">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E4:14"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="binop_1.html#T23" title="BINOP_1:th.23">BINOP_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">B'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">[;]</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">[;]</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:14_2"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="funcop_1.html#T66" title="FUNCOP_1:th.66">FUNCOP_1:66</a></i>
<br/>.= 
<font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B'</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">c</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">[;]</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">f</font></span>)</span>
<b>by </b><i><a class="txt" href="setwop_2.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="setwop_2.html#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="setwop_2.html#E2:14_2"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="setwop_2.html#T4" target="_self" title="SETWOP_2:th.4">Th4</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:14"/>
for <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">C</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S2" title="SETWISEO:sch.2">SETWISEO:sch 2</a>(<a class="txt" href="setwop_2.html#E7:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="setwop_2.html#E8:14"><i><font color="Green" title="E14">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E10:14"/>
<font color="Maroon" title="c7">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c3">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">[;]</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">f</font></span>)</span>
 ;<br/>


</div>
