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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <a href="binop_2.html#K29">multcomplex</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <a href="binop_2.html#K29">multcomplex</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcop_1.html#K8">[;]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="complsp2.html#E2"><i><font color="Green">Lemma2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcop_1.html#K8">[;]</a> <span class="p2">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D3">BINOP_2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="funcop_1.html#K6">.:</a> <span class="p2">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="complsp1.html#T15">COMPLSP1:15</a>, <a class="ref" href="finseqop.html#T36">FINSEQOP:36</a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="finseqop.html#K1">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T31">FUNCOP_1:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complsp1.html#D7">COMPLSP1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K2">COMPLEX</a> </span>)</span></span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="complsp2.html#E2"><i><font color="Green">Lemma2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complsp1.html#K6">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="complsp2.html#E2"><i><font color="Green">Lemma2</font></i></a></i>
;<br/>


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