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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="prvect_1.html#NM1">Domain-Sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="prvect_1.html#M1" target="_self">BinOps</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13">BINOP_1:def 13</a><br/>

<a NAME="E2:47"/><i><font color="Green">E29</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="card_3.html#T18">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">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:47_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="prvect_1.html#E1:47"><i><font color="Green">E28</font></i></a>, <a class="ref" href="prvect_1.html#D10" target="_self">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" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="prvect_1.html#K13" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="binop_1.html#D2">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" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="prvect_1.html#E2:47"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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