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

<b>let </b><font color="Maroon">l</font> be    <a href="csspace.html#L1" target="_self">CUNITSTR</a> ;<br/>

<b>assume </b><a NAME="E1:115"/><i><font color="Green">E31</font></i>: 
 <a href="clvect_1.html#G1">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">l</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">l</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">l</font>,the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">l</font> #) is   <a href="clvect_1.html#NM2">ComplexLinearSpace</a>
 ;<br/>

<a NAME="E2:115"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">l</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="csspace.html#T9" target="_self">Th9</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="clvect_1.html#L1">CLSStruct</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">l0</font> =  <a href="clvect_1.html#G1">CLSStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">l</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">l</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">l</font>,the <a href="clvect_1.html#U1">Mult</a> of <font color="Maroon">l</font> #) as   <a href="clvect_1.html#NM2">ComplexLinearSpace</a> <b>by </b><i><a class="ref" href="csspace.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E5:115"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font> holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>, <font color="Maroon">w</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>;<br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font> ;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font> ;<br/>
<b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font>

<br/>.= 
<font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:115"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font> holds  <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>, <font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>, <font color="Maroon">w</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>;<br/>

<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font> ;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font> ;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font>

<br/>.= 
<font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E7:115"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font> holds  <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>;<br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font> ;<br/>
<b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">l</font></span>)</span> = 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">l0</font></span>)</span>

<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E8:115"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>  ex <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font> st <font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>;<br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font> ;<br/>
<b>consider </b><font color="Maroon">w1</font> being   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font><b> such that </b><br/><a NAME="E3:115_4"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">l0</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">w1</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font> ;<br/>
<a NAME="E5:115_4"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="csspace.html#T12" target="_self">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>thus </b><a NAME="E6:115_4"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:115"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">z</font> being  <a href="cfunct_1.html#NM1">Complex</a><br/> for <font color="Olive">v</font>, <font color="Olive">w</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font> holds  <font color="Olive">z</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">w</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="clvect_1.html#K1">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">z</font> <a href="clvect_1.html#K1">*</a> <font color="Olive">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>, <font color="Maroon">w</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>;<br/>



<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font> ;<br/>
<b>thus </b><font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span> = 
<font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">w</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E10:115"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">z1</font>, <font color="Olive">z2</font> being  <a href="cfunct_1.html#NM1">Complex</a><br/> for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font> holds  <span class="p1">(<span class="default"><font color="Olive">z1</font> <a href="complex1.html#K8">+</a> <font color="Olive">z2</font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">z1</font> <a href="clvect_1.html#K1">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">z2</font> <a href="clvect_1.html#K1">*</a> <font color="Olive">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z1</font> be   <a href="cfunct_1.html#NM1">Complex</a>, <font color="Maroon">z2</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>;<br/>



<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="complex1.html#K8">+</a> <font color="Maroon">z2</font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="complex1.html#K8">+</a> <font color="Maroon">z2</font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v1</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E11:115"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">z1</font>, <font color="Olive">z2</font> being  <a href="cfunct_1.html#NM1">Complex</a><br/> for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font> holds  <span class="p1">(<span class="default"><font color="Olive">z1</font> <a href="complex1.html#K9">*</a> <font color="Olive">z2</font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">z1</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">z2</font> <a href="clvect_1.html#K1">*</a> <font color="Olive">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z1</font> be   <a href="cfunct_1.html#NM1">Complex</a>, <font color="Maroon">z2</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>;<br/>



<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="complex1.html#K9">*</a> <font color="Maroon">z2</font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="complex1.html#K9">*</a> <font color="Maroon">z2</font></span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v1</font>

<br/>.= 
<font color="Maroon">z1</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
<br/>.= 
<font color="Maroon">z1</font> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<a NAME="E12:115"/>
for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font> holds  <a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l</font>;<br/>

<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">l0</font> ;<br/>
<b>thus </b><a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <font color="Maroon">v</font> = 
<a href="complex1.html#K6">1r</a>  <a href="clvect_1.html#K1">*</a> <font color="Maroon">v1</font>

<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="clvect_1.html#D2">CLVECT_1:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:115"/>
<font color="Maroon">l</font> is   <a href="clvect_1.html#NM2">ComplexLinearSpace</a>
 <b>by </b><i><a class="ref" href="csspace.html#T10" target="_self">Th10</a>, , , <a class="ref" href="csspace.html#T11" target="_self">Th11</a>, <a class="ref" href="csspace.html#T12" target="_self">Th12</a>, , <a class="ref" href="csspace.html#D8" target="_self">Def8</a>, <a class="ref" href="clvect_1.html#T1">CLVECT_1:1</a></i>;<br/>


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