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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V2" title="BINOP_1:attr.2">associative</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#V2" title="BINOP_1:attr.2">associative</a> </span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">o</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</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#V2" title="BINOP_1:attr.2">associative</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:79"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c2">o</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</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="c2">o</font> <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D14" title="BINOP_1:def.14">BINOP_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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>,<span class="p1">(<span class="default"><span class="p2">(<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="c5">z</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>,<font color="Maroon" title="c5">z</font></span><br/>

<span class="kw">thus </span><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> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>,<font color="Maroon" title="c5">z</font> = 
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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></span>)</span>,<font color="Maroon" title="c5">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#T45" target="_self" title="MONOID_1:th.45">Th45</a></span>
<br/>.= 
<font color="Maroon" title="c2">o</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p3"><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></span>)</span>,<font color="Maroon" title="c5">z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#T45" target="_self" title="MONOID_1:th.45">Th45</a></span>
<br/>.= 
<font color="Maroon" title="c2">o</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p3"><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="c5">z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="monoid_1.html#T49" target="_self" title="MONOID_1:th.49">Th49</a></span>
<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="c3">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><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="c5">z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#T45" target="_self" title="MONOID_1:th.45">Th45</a></span>
<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="c3">x</font>,<span class="p1">(<span class="default"><span class="p2">(<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="c5">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#T45" target="_self" title="MONOID_1:th.45">Th45</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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