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



<b>assume </b><a NAME="E1:30"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="binop_1.html#V1" target="_self">commutative</a>
 ;<br/>

<a NAME="E2:30"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#R6" target="_self">is_distributive_wrt</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#R4" target="_self">is_left_distributive_wrt</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="binop_1.html#T26" target="_self">Th26</a></i>;<br/>
<b>hence </b><a NAME="E3:30"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#R5" target="_self">is_right_distributive_wrt</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#R4" target="_self">is_left_distributive_wrt</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="binop_1.html#E1:30"><i><font color="Green">E21</font></i></a>, <a class="ref" href="binop_1.html#T27" target="_self">Th27</a></i>;<br/>


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