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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">F</font>, <font color="Olive" title="b3">G</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">F</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b3">G</font> holds <br/><font color="Olive" title="b2">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Olive" title="b1">RD</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b3">G</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Olive" title="b1">RD</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</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">F</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b2">G</font> holds <br/><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> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b2">G</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font>, <font color="Maroon" title="c4">G</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">G</font> implies <font color="Maroon" title="c3">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">G</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c4">G</font>
 <span class="kw">and </span><br/><a NAME="E2:19"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c4">G</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D11" title="BINOP_1:def.11">BINOP_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c4">G</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font></span><br/>

<span class="kw">thus </span><a NAME="E3:19"/>
( <font color="Maroon" title="c3">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c4">G</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font> &amp; <font color="Maroon" title="c3">F</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c4">G</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <font color="Maroon" title="c2">RD</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="filter_1.html#T9" target="_self" title="FILTER_1:th.9">Th9</a>, <a class="ref" href="filter_1.html#T10" target="_self" title="FILTER_1:th.10">Th10</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D11" title="BINOP_1:def.11">BINOP_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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