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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E10">Th12</font></span>: <a NAME="T12"><span class="comment"><font color="firebrick">:: BINOP_1:12</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">o</font>, <font color="Olive" title="b3">o9</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">A</font>  st <font color="Olive" title="b3">o9</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  holds <br/>( <font color="Olive" title="b3">o9</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> <font color="Olive" title="b2">o</font> iff  for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">A</font> holds  <font color="Olive" title="b3">o9</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b4">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">o9</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">o9</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b4">a</font>,<font color="Olive" title="b6">c</font>)</span>)</span>) )</div></div>
