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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b4">g1</font>, <font color="Olive" title="b5">g2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b2">f1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b3">f2</font> holds <br/><font color="Olive" title="b4">g1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b5">g2</font></span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b3">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b1">f1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b2">f2</font> holds <br/><font color="Olive" title="b3">g1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b4">g2</font></span><br/><b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <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">f2</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b2">g2</font></span><br/><b>let </b><font color="Maroon" title="c5">g1</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">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c6">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <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">f2</font> implies <font color="Maroon" title="c5">g1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c6">g2</font> )</span><br/>







<b>assume </b><a NAME="E1:4"/>
( <font color="Maroon" title="c5">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c6">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">S</font> &amp; <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">f2</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">f2</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">g1</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c6">g2</font></span><br/>

<b>hence </b><a NAME="E2:4"/>
( <font color="Maroon" title="c5">g1</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">g1</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_2.html#T3" target="_self" title="FILTER_2:th.3">Th3</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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