<?xml version="1.0"?>
<div class="add">

<b>thus </b><a NAME="E1:7_1_1"/>
 <a href="complfld.html#K1" target="_self">F_Complex</a>  is <a href="rlvect_1.html#V4">add-associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</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="complfld.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<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>,<font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</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"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D3">BINOP_2:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D3">BINOP_2:def 3</a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D3">BINOP_2:def 3</a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<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>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:7_1_1"/>
 <a href="complfld.html#K1" target="_self">F_Complex</a>  is <a href="rlvect_1.html#V5">right_zeroed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</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="complfld.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1" target="_self">F_Complex</a> </span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1" target="_self">F_Complex</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1" target="_self">F_Complex</a> </span>)</span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> 

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<a href="complex1.html#K5">0c</a> 
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<a href="complex1.html#K5">0c</a> 
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K8">+</a> <a href="complex1.html#K5">0c</a> 
<b>by </b><i><a class="ref" href="binop_2.html#D3">BINOP_2:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="complex1.html#D6">COMPLEX1:def 6</a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:7_1_1"/>
 <a href="complfld.html#K1" target="_self">F_Complex</a>  is <a href="rlvect_1.html#V6">right_complementable</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</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="complfld.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>

<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D3">BINOP_2:def 3</a></i>
<br/>.= 
the <a href="struct_0.html#U2">Zero</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> 
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a>, <a class="ref" href="xcmplx_0.html#D6">XCMPLX_0:def 6</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1" target="_self">F_Complex</a> 

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:7_1_1"/>
 <a href="complfld.html#K1" target="_self">F_Complex</a>  is <a href="rlvect_1.html#V3">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_4"><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="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</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="complfld.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <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="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D3">BINOP_2:def 3</a></i>
<br/>.= 
<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>3</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D3">BINOP_2:def 3</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>1</sub></font>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:7_1_1"/>
 <a href="complfld.html#K1" target="_self">F_Complex</a>  is <a href="group_1.html#V7">commutative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_5"><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="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16">GROUP_1:def 16</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</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="complfld.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <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="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <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="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:7_1_1"/>
 <a href="complfld.html#K1" target="_self">F_Complex</a>  is <a href="group_1.html#V4">associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_6"><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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</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="complfld.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>

<br/>.= 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</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>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:7_1_1"/>
 <a href="complfld.html#K1" target="_self">F_Complex</a>  is <a href="vectsp_1.html#V8">left_unital</a>
 ;<br/>

<b>thus </b><a NAME="E8:7_1_1"/>
 <a href="complfld.html#K1" target="_self">F_Complex</a>  is <a href="vectsp_1.html#V6">right_unital</a>
 ;<br/>

<b>thus </b><a NAME="E9:7_1_1"/>
 <a href="complfld.html#K1" target="_self">F_Complex</a>  is <a href="vectsp_1.html#V7">distributive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_7"><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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</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="complfld.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

<br/>.= 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>

<br/>.= 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D3">BINOP_2:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D3">BINOP_2:def 3</a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</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="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</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="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</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="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</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">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>

;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K8">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D3">BINOP_2:def 3</a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="complex1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E10:7_1_1"/>
 <a href="complfld.html#K1" target="_self">F_Complex</a>  is <a href="vectsp_1.html#V9">Field-like</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D20">VECTSP_1:def 20</a><br/>

<b>assume </b><a NAME="E1:7_1_1_8"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1" target="_self">F_Complex</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</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="complfld.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K12">"</a>  as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<a NAME="E4:7_1_1_8"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U2">Zero</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> 
 
<b>by </b><i><a class="txt" href="complfld.html#E1:7_1_1_8"><i><font color="Green">E4</font></i></a></i>;<br/>
<a NAME="E5:7_1_1_8"/><b>then </b><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> = 
the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K12">"</a> </span>)</span>

<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <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>2</sub></font> <a href="complex1.html#K12">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complex1.html#K12">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
<br/>.= 
 <a href="complex1.html#K6">1r</a> 
<b>by </b><i><a class="txt" href="complfld.html#E5:7_1_1_8"><i><font color="Green">E5</font></i></a>, <a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1" target="_self">F_Complex</a> 
<b>by </b><i><a class="txt" href="complfld.html#E5"><i><font color="Green">Lemma3</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E11:7_1_1"/>
not  <a href="complfld.html#K1" target="_self">F_Complex</a>  is <a href="vectsp_1.html#V10">degenerated</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_9"><b>proof </b></a><div class="add">

 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1" target="_self">F_Complex</a>  = 
the <a href="struct_0.html#U2">Zero</a> of <a href="complfld.html#K1" target="_self">F_Complex</a> 

<br/>.= 
 <a href="complex1.html#K5">0c</a> 
<b>by </b><i><a class="ref" href="complfld.html#D1" target="_self">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E2:7_1_1_9"/>
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1" target="_self">F_Complex</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1" target="_self">F_Complex</a> 
 <b>by </b><i><a class="txt" href="complfld.html#E5"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="complex1.html#T12">COMPLEX1:12</a>, <a class="ref" href="complex1.html#T15">COMPLEX1:15</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a><br/>


</div><b>end;</b></div>


</div>
