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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ] means <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E1:4"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ;<br/>

<a NAME="E1:4_1"/>
<span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is    <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> 
 
<b>by </b><i><a class="ref" href="csspace.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E2:4_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="binop_1.html#NM2">BinOp</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a>  st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="csspace.html#E1:4"><i><font color="Green">E5</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> <b> such that </b><br/><a NAME="E4:4"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a>  holds <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 ;<br/>
<a NAME="E5:4"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ;<br/><b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace.html#E4:4"><i><font color="Green">E6</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="csspace.html#E4:4"><i><font color="Green">E6</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:4_2"/>
<font color="Maroon">c<sub>1</sub></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/>


</div><b>end;</b></div>
<a NAME="E6:4"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <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="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a> ;<br/><i><font color="Green">E8</font></i>:  <a href="csspace.html#K2" target="_self">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="csspace.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace.html#E4:4"><i><font color="Green">E6</font></i></a></i>
;<br/><i><font color="Green">E9</font></i>: <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="csspace.html#E4:4"><i><font color="Green">E6</font></i></a></i>
<br/>.= 
 <a href="csspace.html#K2" target="_self">seq_id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D2" target="_self">Def2</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="csspace.html#E4:4"><i><font color="Green">E6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p2">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="csspace.html#E1:4_3_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="comseq_1.html#T9">COMSEQ_1:9</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="csspace.html#E4:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="csspace.html#E2:4_3_1"><i><font color="Green">E9</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:4_3"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a>
 <b>by </b><i><a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="binop_1.html#NM2">BinOp</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a>  st <br/>( ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="csspace.html#K1" target="_self">the_set_of_ComplexSequences</a>  holds <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" target="_self">seq_id</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) &amp; <font color="Olive">b<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a> )
 <b>by </b><i><a class="txt" href="csspace.html#E4:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="csspace.html#E5:4"><i><font color="Green">E7</font></i></a></i>;<br/>


</div>
