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<b>let </b><font color="Maroon" title="c1">D</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> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">o</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> holds <br/><font color="Olive" title="b1">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a></span><br/><b>let </b><font color="Maroon" title="c2">o</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">o</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> implies <font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> )</span><br/>



<b>assume </b><a NAME="E1:78"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c2">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="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a></span><br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> = 
<font color="Maroon" title="c2">o</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="monoid_1.html#T45" target="_self" title="MONOID_1:th.45">Th45</a></i>
<br/>.= 
<font color="Maroon" title="c2">o</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E1:78"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="monoid_1.html#T48" target="_self" title="MONOID_1:th.48">Th48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="monoid_1.html#T45" target="_self" title="MONOID_1:th.45">Th45</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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