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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="prvect_1.html#NM1" title="PRVECT_1:NM.1">Domain-Sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">b</font> be    <a href="prvect_1.html#M1" title="PRVECT_1:mode.1">BinOps</a> of <font color="Maroon" title="c1">a</font>;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E28">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font> holds  <font color="Maroon" title="c2">b</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Olive" title="b1">i</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">a</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><br/>

<a NAME="E2:47"/><i><font color="Green" title="E29">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c5">z</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font> ;<br/><a NAME="E3:47_1"/>
( <font color="Maroon" title="c2">b</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c6">i</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="prvect_1.html#E1:47"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="prvect_1.html#D10" target="_self" title="PRVECT_1:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E4:47_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:47"/>
<span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></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"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></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>
 <b>by </b><i><a class="txt" href="prvect_1.html#E2:47"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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