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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>











<b>thus </b><a NAME="E1:44"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">c<sub>6</sub></font> implies <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#R6">is_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K6" target="_self">:|</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#R4">is_left_distributive_wrt</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#R5">is_right_distributive_wrt</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#R4">is_left_distributive_wrt</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#R5">is_right_distributive_wrt</a> <font color="Maroon">c<sub>6</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D11">BINOP_1:def 11</a><br/>

<b>hence </b><a NAME="E2:44_1"/>
( <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#R4">is_left_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K6" target="_self">:|</a></span> &amp; <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#R5">is_right_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K6" target="_self">:|</a></span> )
 <b>by </b><i><a class="ref" href="filter_1.html#T28" target="_self">Th28</a>, <a class="ref" href="filter_1.html#T29" target="_self">Th29</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D11">BINOP_1:def 11</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:44"/>
( <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#R4">is_left_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K6" target="_self">:|</a></span> &amp; <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="filter_1.html#K6" target="_self">:|</a></span> <a href="binop_1.html#R5">is_right_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="filter_1.html#K6" target="_self">:|</a></span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D11">BINOP_1:def 11</a><br/>

<b>hence </b><a NAME="E3:44"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#R4">is_left_distributive_wrt</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#R5">is_right_distributive_wrt</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#R4">is_left_distributive_wrt</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#R5">is_right_distributive_wrt</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="filter_1.html#T28" target="_self">Th28</a>, <a class="ref" href="filter_1.html#T29" target="_self">Th29</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D11">BINOP_1:def 11</a><br/>


</div>
