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<span class="kw">let </span><font color="Maroon" title="c1">A</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">o</font>, <font color="Olive" title="b2">o9</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">o9</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  holds <br/>( <font color="Olive" title="b2">o9</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Olive" title="b1">o</font> iff <font color="Olive" title="b2">o9</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Olive" title="b1">o</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">o9</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">A</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">o9</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  implies ( <font color="Maroon" title="c3">o9</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c2">o</font> iff <font color="Maroon" title="c3">o9</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c2">o</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c3">o9</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</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">o9</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c2">o</font> iff <font color="Maroon" title="c3">o9</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c2">o</font> )</span><br/>

<a NAME="E2:42"/><span class="kw">then </span>
( <font color="Maroon" title="c3">o9</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Maroon" title="c2">o</font> iff <font color="Maroon" title="c3">o9</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c2">o</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T26" target="_self" title="BINOP_1:th.26">Th26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:42"/>
( <font color="Maroon" title="c3">o9</font> <a href="binop_1.html#R5" title="BINOP_1:pred.5">is_right_distributive_wrt</a> <font color="Maroon" title="c2">o</font> iff <font color="Maroon" title="c3">o9</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c2">o</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="binop_1.html#T27" target="_self" title="BINOP_1:th.27">Th27</a></span>; <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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