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

<b>let </b><font color="Maroon">D1</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">D2</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">f1</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">g1</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D1</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D2</font>;<br/><b>let </b><font color="Maroon">g2</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D2</font>;<br/>











<b>thus </b><a NAME="E1:44"/>
( <font color="Maroon">f1</font> <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">f2</font> <a href="binop_1.html#R6">is_distributive_wrt</a> <font color="Maroon">g2</font> implies <span class="p1"><a href="filter_1.html#K6" target="_self">|:</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</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">g1</font>,<font color="Maroon">g2</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">f1</font> <a href="binop_1.html#R4">is_left_distributive_wrt</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">f1</font> <a href="binop_1.html#R5">is_right_distributive_wrt</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">f2</font> <a href="binop_1.html#R4">is_left_distributive_wrt</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">f2</font> <a href="binop_1.html#R5">is_right_distributive_wrt</a> <font color="Maroon">g2</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">f1</font>,<font color="Maroon">f2</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">g1</font>,<font color="Maroon">g2</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">f1</font>,<font color="Maroon">f2</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">g1</font>,<font color="Maroon">g2</font></span><a href="filter_1.html#K6" target="_self">:|</a></span> )
 <b>by </b><i>, </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">f1</font>,<font color="Maroon">f2</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">g1</font>,<font color="Maroon">g2</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">f1</font>,<font color="Maroon">f2</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">g1</font>,<font color="Maroon">g2</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">f1</font> <a href="binop_1.html#R4">is_left_distributive_wrt</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">f1</font> <a href="binop_1.html#R5">is_right_distributive_wrt</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">f2</font> <a href="binop_1.html#R4">is_left_distributive_wrt</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">f2</font> <a href="binop_1.html#R5">is_right_distributive_wrt</a> <font color="Maroon">g2</font> )
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D11">BINOP_1:def 11</a><br/>


</div>
