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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">D</font> being 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> <br/> for <font color="Olive" title="b2">B</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Olive" title="b1">D</font>  st <font color="Olive" title="b2">B</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> holds <br/> <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Olive" title="b2">B</font>,<font color="Maroon" title="c1">n</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a></span><br/><b>let </b><font color="Maroon" title="c2">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">B</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">B</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> holds <br/> <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Olive" title="b1">B</font>,<font color="Maroon" title="c1">n</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a></span><br/><b>let </b><font color="Maroon" title="c3">B</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">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="c3">B</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> implies  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )</span><br/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c3">B</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">D</font>; <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"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></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"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span></span><br/><b>thus </b><span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></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"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font> = 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c3">B</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span>,<font color="Maroon" title="c6">z</font>
<b>by </b><i><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c3">B</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="txt" href="prvect_1.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="finseqop.html#T29" title="FINSEQOP:th.29">FINSEQOP:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="prvect_1.html#K1" title="PRVECT_1:func.1">.:</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:17"/>
 <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c1">n</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</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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