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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>





<span class="kw">assume </span><a NAME="E1:25"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="25_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="finseqop.html#T29" title="FINSEQOP:th.29">FINSEQOP:29</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_1.html#D1" target="_self" title="PRVECT_1:def.1">Def1</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:25"/>
 <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 class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</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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