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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">o'</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">o'</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> holds <br/>( <font color="Olive" title="b2">o'</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">o'</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/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">o'</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">o'</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> implies ( <font color="Maroon" title="c3">o'</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">o'</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/>



<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c3">o'</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">o'</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">o'</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="E2:30"/><b>then </b>
( <font color="Maroon" title="c3">o'</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">o'</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c2">o</font> )
 
<b>by </b><i><a class="ref" href="binop_1.html#T26" target="_self" title="BINOP_1:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E3:30"/>
( <font color="Maroon" title="c3">o'</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">o'</font> <a href="binop_1.html#R4" title="BINOP_1:pred.4">is_left_distributive_wrt</a> <font color="Maroon" title="c2">o</font> )
 <b>by </b><i><a class="txt" href="binop_1.html#E1:30"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="binop_1.html#T27" target="_self" title="BINOP_1:th.27">Th27</a></i>; <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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