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



<b>assume </b><a NAME="E1:38"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>set </b><font color="Maroon">cM</font> = <font color="Maroon">c</font> <a href="complsp1.html#K1">multcomplex</a> ;<br/>
<a NAME="E4:38"/>
<font color="Maroon">c</font> is    <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/>
<a NAME="E5:38"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c</font> <a href="complsp1.html#K1">multcomplex</a>  <a href="binop_1.html#R7">is_distributive_wrt</a>  <a href="binop_2.html#K27">addcomplex</a> 
 
<b>by </b><i><a class="ref" href="complsp1.html#T17">COMPLSP1:17</a></i>;<br/>
<font color="Maroon">c</font> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K3">+</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K1">multcomplex</a> </span>)</span> <a href="finseqop.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K3">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp1.html#D10">COMPLSP1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K1">multcomplex</a> </span>)</span> <a href="finseqop.html#K5">*</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp1.html#D7">COMPLSP1:def 7</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"><font color="Maroon">c</font> <a href="complsp1.html#K1">multcomplex</a> </span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">x'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K1">multcomplex</a> </span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseqop.html#T52">FINSEQOP:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K1">multcomplex</a> </span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">x</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K1">multcomplex</a> </span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">y</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</font> <a href="complsp1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K1">multcomplex</a> </span>)</span> <a href="finseqop.html#K5">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp1.html#D10">COMPLSP1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K6">*</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp1.html#D10">COMPLSP1:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E7:38"/>
<font color="Maroon">c</font> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K3">+</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K6">*</a> <font color="Maroon">y</font></span>)</span>
 ;<br/>


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