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<b>let </b><font color="Maroon" title="c1">D1</font>, <font color="Maroon" title="c2">D2</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">f1</font>, <font color="Maroon" title="c4">g1</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</font>;<br/><b>let </b><font color="Maroon" title="c5">f2</font>, <font color="Maroon" title="c6">g2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font>;<br/>











<b>thus </b><a NAME="E1:44"/>
( <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> implies <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</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" title="c3">f1</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D11" title="BINOP_1:def.11">BINOP_1:def 11</a><br/>

<b>hence </b><a NAME="E2:44_1"/>
( <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> &amp; <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )
 <b>by </b><i><a class="ref" href="filter_1.html#T28" target="_self" title="FILTER_1:th.28">Th28</a>, <a class="ref" href="filter_1.html#T29" target="_self" title="FILTER_1:th.29">Th29</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D11" title="BINOP_1:def.11">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" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> &amp; <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c5">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c4">g1</font>,<font color="Maroon" title="c6">g2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D11" title="BINOP_1:def.11">BINOP_1:def 11</a><br/>

<b>hence </b><a NAME="E3:44"/>
( <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c3">f1</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> &amp; <font color="Maroon" title="c5">f2</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> )
 <b>by </b><i><a class="ref" href="filter_1.html#T28" target="_self" title="FILTER_1:th.28">Th28</a>, <a class="ref" href="filter_1.html#T29" target="_self" title="FILTER_1:th.29">Th29</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D11" title="BINOP_1:def.11">BINOP_1:def 11</a><br/>


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