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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/>



<a NAME="E1:28"/>
( <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> iff for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="binop_1.html#D9" target="_self">Def9</a></i>;<br/>
<b>hence </b><a NAME="E2:28"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="binop_1.html#V1" target="_self">commutative</a> implies ( <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#T24" target="_self">Th24</a></i>;<br/>


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