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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>





<a NAME="E1:16"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:16"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">B</font> is <a href="binop_1.html#V1">commutative</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>, <font color="Maroon">y</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">D</font>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> <font color="Maroon">B</font>,<font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">B</font> <a href="prvect_1.html#K1" target="_self">.:</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">B</font> <a href="prvect_1.html#K1" target="_self">.:</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="finseqop.html#T34">FINSEQOP:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> <font color="Maroon">B</font>,<font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
 <a href="prvect_1.html#K2" target="_self">product</a> <font color="Maroon">B</font>,<font color="Maroon">n</font> is <a href="binop_1.html#V1">commutative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>


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